Analytic Theory of Global Bifurcation [1 ed.] 0691112983

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Analytic Theory of Global Bifurcation

PRINCETON SERIES IN APPLIED MATHEMATICS

EDITORS Ingrid Daubechies, Princeton University Weinan E, Princeton University Jan Karel Lenstra, Georgia Institute of Technology Endre S¨uli, University of Oxford

TITLES IN THE SERIES Chaotic Transitions in Deterministic and Stochastic Dynamical Systems: Applications of Melnikov Processes in Engineering, Physics and Neuroscience by Emil Simiu Selfsimilar Processes by Paul Embrechts and Makoto Maejima Self-Regularity: A new Paradigm for Primal-Dual Interior Points Algorithms by Jiming Peng, Cornelis Roos and Tam´as Terlaky Analytic Theory of Global Bifurcation: An Introduction by Boris Buffoni and John Toland The Princeton Series in Applied Mathematics publishes high quality advanced texts and monographs in all areas of applied mathematics. Books include those of a theoretical and general nature as well as those dealing with the mathematics of specific applications areas and real-world situations.

Analytic Theory of Global Bifurcation

An Introduction

Boris Buffoni and John Toland

PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD

c 2003 by Princeton University Press Copyright
Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 3 Market Place, Woodstock, Oxfordshire OX20 1SY All Rights Reserved ISBN 0-691-11298-3 (cloth : alk. paper) British Library Cataloging-in-Publication Data has been applied for This book has been composed in Times by Stephen I. Pargeter, Banbury, Oxfordshire, UK Printed on acid-free paper. ∞
www.pupress.princeton.edu Printed in the United States of America 10

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1

Contents

Preface

ix

Chapter 1. Introduction

1

1.1 1.2 1.3 1.4

Example: Bending an Elastic Rod I Principle of Linearization Global Theory Layout

2 5 6 7

PART 1. LINEAR AND NONLINEAR FUNCTIONAL ANALYSIS

9

Chapter 2. Linear Functional Analysis

11

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

Preliminaries and Notation Subspaces Dual Spaces Linear Operators Neumann Series Projections and Subspaces Compact and Fredholm Operators Notes on Sources

Chapter 3. Calculus in Banach Spaces 3.1 3.2 3.3 3.4 3.5 3.6 3.7

Fr´echet Differentiation Higher Derivatives Taylor’s Theorem Gradient Operators Inverse and Implicit Function Theorems Perturbation of a Simple Eigenvalue Notes on Sources

Chapter 4. Multilinear and Analytic Operators 4.1 4.2 4.3 4.4

Bounded Multilinear Operators Fa`a de Bruno Formula Analytic Operators Analytic Functions of Two Variables

11 13 14 15 16 17 18 20 21 21 27 31 32 35 38 40 41 41 44 45 52

vi

CONTENTS

4.5 4.6

Analytic Inverse and Implicit Function Theorems Notes on Sources

PART 2.

ANALYTIC VARIETIES

Chapter 5. Analytic Functions on Fn 5.1 5.2 5.3 5.4 5.5

Preliminaries Weierstrass Division Theorem Weierstrass Preparation Theorem Riemann Extension Theorem Notes on Sources

Chapter 6. Polynomials 6.1 6.2 6.3

Constant Coefficients Variable Coefficients Notes on Sources

Chapter 7. Analytic Varieties 7.1 7.2 7.3 7.4 7.5 7.6

F-Analytic Varieties Weierstrass Analytic Varieties Analytic Germs and Subspaces Germs of C-analytic Varieties One-dimensional Branches Notes on Sources

PART 3.

53 57

59 61 61 64 65 66 69 70 70 74 77 78 78 81 86 88 95 99

BIFURCATION THEORY

101

Chapter 8. Local Bifurcation Theory

103

8.1 8.2 8.3 8.4 8.5 8.6 8.7

A Necessary Condition Lyapunov-Schmidt Reduction Crandall-Rabinowitz Transversality Bifurcation from a Simple Eigenvalue Bending an Elastic Rod II Bifurcation of Periodic Solutions Notes on Sources

Chapter 9. Global Bifurcation Theory 9.1 9.2 9.3 9.4

Global One-Dimensional Branches Global Analytic Bifurcation in Cones Bending an Elastic Rod III Notes on Sources

103 104 105 109 111 112 113 114 114 120 121 124

vii

CONTENTS

PART 4.

STOKES WAVES

Chapter 10. Steady Periodic Water Waves 10.1 10.2 10.3 10.4 10.5 10.6

Euler Equations One-dimensional Formulation Main Equation A Priori Bounds and Nekrasov’s Equation Weak Solutions Are Classical Notes on Sources

Chapter 11. Global Existence of Stokes Waves 11.1 11.2 11.3 11.4

Local Bifurcation Theory Global Bifurcation from λ = 1 Gradients, Morse Index and Bifurcation Notes on Sources

125 127 127 131 137 140 146 151 152 152 154 157 159

Bibliography

161

Index

167

Preface

In the study of homogeneous linear problems of the form Lu = 0, the amplitude (or size) of a solution is unimportant as non-zero solutions are never isolated and the existence of one non-zero solution implies the existence of others with all possible amplitudes. This is not the case for nonlinear equations of the form N (u) = 0 and the observation that a solution is unique in a neighbourhood of itself has no ramifications globally for the set of all solutions of the equation. Since nonlinear equations are often derived without restrictions on the amplitudes of likely solutions, with the intention of describing large amplitude phenomena, it is essential to have mathematical methods which discover the existence of solutions without regard to their size. These notes are concerned with analytical aspects of this general question. Global bifurcation theory deals with the existence in-the-large of connected sets of solutions to nonlinear equations of the form F (λ, x) = 0, λ ∈ R, x ∈ X \ {0}, where F : R×X → Y , X, Y are Banach spaces and F (λ, 0) ≡ 0. It is well known that P. H. Rabinowitz’s now-classical topological theory of global bifurcation leads to the existence of sets of solutions which are connected, but not in general pathconnected even when the operators involved are infinitely differentiable. On the other hand, E. N. Dancer pointed out that connectedness in the topological theory can be replaced with path-connectedness if the operators are real-analytic. The two approaches from the early 1970s are completely different. Quite recently, in a collaboration with Dancer on a problem from hydrodynamic wave theory, we encountered the following situation. If the existence globally of a path of solutions of a certain real-analytic equation could be established, it was clear from earlier work of P. I. Plotnikov that along the path the Morse index must increase without bound; this in turn would lead to infinitely many secondary bifurcation points and ultimately to the existence of sub-harmonic bifurcations, which was our goal. Here, as in many similar problems, the existence globally of paths (as opposed to connected sets) of solutions is the key. Our purpose therefore is to give a selfcontained account of such a theory, and in particular to focus on the existence of global paths of solutions as a consequence of bifurcation from a simple eigenvalue in the real-analytic case. There follows an expanded version of the notes for a course of postgraduate lectures on real-analytic global bifurcation theory and its applications which we

x

PREFACE

gave as part of ‘un cour de 3`eme cycle a` l’Ecole Polytechnique F´ed´erale de Lausanne’ during the winter term of the academic year 1999-2000. We have benefited greatly from the interest of those who attended the lectures, asked questions, read drafts and offered advice: in particular we thank A. Andr´e, D. Crispin, A. PichlerTennenberg, F. Gebran, S. Rey and C. A. Stuart. We must also thank E. N. Dancer for his encouragement and comments on the manuscript. We assume a knowledge of undergraduate linear functional analysis and of calculus notation in finite dimensions. We also assume a knowledge of a first course in functions of one complex variable and some elementary linear algebra. However calculus in Banach spaces is treated with complete proofs, from the definition of a Fr´echet derivative to the inverse and implicit function theorems. We include an account of how infinite-dimensional problems can be reduced locally to equivalent ones in finite dimensions and of how that leads to standard results from local bifurcation theory. This theory assumes only that operators have a specified finite degree of differentiability. Then we develop the theory of infinite-dimensional analytic operators over R or C and re-prove the inverse and implicit function theorems in that context. The elementary parts of the theory of finite-dimensional analytic varieties are also developed from first principles. Finally, we study the global theory of real-analytic, one-dimensional continua of solutions. This has been the main goal, but the methods and results discussed here have much wider applicability to nonlinear operator equations. The application to the water-wave problem is considered in some detail. A list of notation is included in the index. Acknowledgements: Boris Buffoni held a grant from the Swiss National Science Foundation and John Toland was a Senior Fellow supported by the UK’s EPSRC during the preparation of this manuscript.

Boris Buffoni, Lausanne John Toland, Bath 31 March 2002

Chapter One Introduction Consider a system of k scalar equations in the form F (λ, x) = 0 ∈ Fk ,

(1.1)

where x ∈ Fn represents the state of a system and λ ∈ Fm is a vector parameter which controls x. (Here F denotes the real or complex field.) A solution of (1.1) is a pair (λ, x) ∈ Fm × Fn and the goal is to say as much as possible qualitatively about the solution set. Since (1.1) is a finite-dimensional nonlinear equation it might seem unnecessarily restrictive or even pointless to distinguish between the λ and x variables. Why not instead write (λ, x) = Z ∈ Fm+n and study the equation F (Z) = 0 where singularity theory is all that is needed? For example, when F : Cm+n → Ck is given by a power series expansion (that is, F is analytic), a solution Z0 is called a bifurcation point if, in every neighbourhood of Z0 , the solutions of F (Z) = 0 do not form a smooth manifold. Locally the solutions form an analytic variety, a finite union of analytic manifolds of possibly different dimensions. So the qualitative theory of F (Z) = 0 in complex finite dimensions is reasonably complete. However (i) in our applications λ is a parameter and the dependence on λ of the solution set is important; (ii) we are looking for a theory that gives the existence globally (i.e. not only in a neighbourhood of a point) of connected sets of solutions; (iii) we are particularly interested in the infinite-dimensional equation F (λ, x) = 0

(1.2)

when X and Y are real Banach spaces, F : R × X → Y is real-analytic and F (λ, 0) ≡ 0. Let Sλ = {x ∈ X : F (λ, x) = 0}. The set Sλ normally depends on the choice of λ and usually varies continuously as λ varies. However, it sometimes happens that there is an abrupt change, a bifurcation, in the solution set, as λ passes through a particular point λ0 . For example, in Figure 1.1 the number of solutions changes from one to two as λ increases through λ0 . For a general treatment of bifurcation theory, see [19]. At this stage it is useful to see an infinite-dimensional example in which the global solution set can be found explicitly.

2

CHAPTER 1

Figure 1.1 The set Sλ splits in two as λ passes through λ0 .

Figure 1.2 The rod bends under the action of a force.

1.1 EXAMPLE: BENDING AN ELASTIC ROD I Consider an elastic rod of length L > 0 with one end fixed at the origin of the (x, y)-plane and with the other free to move on the x-axis under the influence of a force along the x-axis towards the origin. If we suppose that the length of the rod does not change (that it is incompressible) and if the force is big enough, then the rod will bend (see Figure 1.2). We suppose that the rod always lies in the (x, y)-plane (there is no twisting out of the plane in the simple model which follows). To describe the rod’s configuration let (x(s), y(s)) be the coordinates of a point at distance s (measured along the rod) from the end which is fixed at the origin. Since
s
s cos φ(t)dt and y(s) = sin φ(t)dt, x(s) = 0

0

the shape of the rod is given by the angle φ(s) between the tangent to the rod and the horizontal at the point (x(s), y(s)), s ∈ [0, L].

3

INTRODUCTION

Figure 1.3 The angle between the tangent and the horizontal.

Let P denote the applied force. Then the Euler-Bernoulli theory [5, 6] of bending says that the curvature of the rod at a point is proportional to the moment created by the force. In other words, −kφ
(s) = P y(s), where k, the constant of proportionality, is determined by the material properties of the rod, P y(s) is the moment of the applied force and −φ
(s) is the curvature at the point (x(s), y(s)). It follows that if P = 0 then φ must be constant, and that constant must be 0 (mod 2π) since y(L) = 0. From now on we consider only the case P > 0. Since y
(s) = sin φ(s) and y(0) = y(L) = 0 this gives φ

(s) + λ sin φ(s) = 0, s ∈ [0, L],

φ
(0) = φ
(L) = 0,

(1.3)

where λ = P/k > 0. If φ is a solution of (1.3), then so is 2kπ + φ, for any k ∈ Z. We therefore assume that φ(0) ∈ (−π, π). (If φ(0) = ±π then φ is a constant.) For all λ > 0, (λ, φ) = (λ, 0) is a solution of (1.3). This means that the mathematical model of bending admits a solution representing a straight rod, irrespective of how large the applied force might be. These solutions, φ = 0, λ > 0 arbitrary, comprise the family of trivial solutions. To be realistic the model must also have solution corresponding to a bent rod (such as depicted in Figures 1.2 and 1.3). Note that any solution of (1.3) must satisfy the identity φ
(s)2 + 4λ sin2 ( 12 φ(s)) = 4λ sin2 ( 12 φ0 ), s ∈ [0, L],

(1.4)

where φ0 = φ(0). This means that (φ(s), φ
(s)), s ∈ [0, L], lies on a segment of the curve in (φ, φ
)-phase space (see Figure 1.4) given implicitly by 

 (φ, φ
) ∈ R2 : φ
2 + 4λ sin2 12 φ = 4λ sin2 12 φ0 ⊂ R2 .

We therefore see that there is a solution joining (−|φ0 |, 0) to (|φ0 |, 0) in the half-space {(φ, φ
) ∈ R2 , φ
≥ 0} and one joining (|φ0 |, 0) to (−|φ0 |, 0) in the

4

CHAPTER 1

Figure 1.4 The direction of solutions in phase space.

half-space {(φ, φ
) ∈ R2 , φ
≤ 0}, the corresponding value of L being given by the formula
L
|φ0 | |dφ| dφ  L= = |dφ|/ds 2 0 −|φ0 | 4λ sin 12 φ0 − 4λ sin2 12 φ
π/2 dθ 1  , =√ λ −π/2 1 − sin2 1 |φ | sin2 θ 0 2 where θ ∈ [−π/2, π/2] is given by sin(φ/2) = sin(|φ0 |/2) sin θ. In fact there are other solutions of (1.4) which in Figure 1.4 go around the curve 12 K times for any positive integer K. For such solutions K L= √ λ




π/2

−π/2

dθ



.

1 − sin2 12 |φ0 | sin2 θ

This integral increases in |φ0 | and converges to +∞ as |φ0 | → π. Since L is the given length of the rod, this relation for each K is an implicit relation between φ0 = φ(0) and λ when (λ, φ) is a solution of (1.3). We can best describe the situation with the aid of a bifurcation diagram in which λ is the horizontal axis, φ0 is the vertical axis, and L is fixed, see Figure 1.5. The different curves correspond to different values of K, and it is easily checked that the K th curve intersects the horizontal axis at (Kπ/L)2 . It is fortunate but unusual that (1.3) can be reduced to (1.4) and that L can be calculated in terms of elliptic integrals. Because of this, solutions to (1.3) of all amplitudes can be found more-or-less explicitly. This is not the case for slightly more complicated problems and almost never for partial differential equations (PDEs).

5

INTRODUCTION

Figure 1.5 Bifurcation diagram.

General methods suitable for PDE applications are based on a study of (1.2). To put (1.3) in such a setting let X = {φ ∈ C 2 [0, L] : φ
(0) = φ
(L) = 0}, Y = C[0, L] and F (λ, φ) = φ

+ λ sin φ ∈ Y, for all (λ, φ) ∈ R × X. Then F : R × X → Y is smooth (Chapter 3) and realanalytic (Chapter 4). In Chapter 8 we show how equation (1.2) can be reduced locally to a finite-dimensional problem. This means that if (λ0 , x0 ) satisfies (1.2) then there is a neighbourhood U of (λ0 , x0 ) in R × X, a neighbourhood V of (λ0 , 0) ∈ R × RN and an equation f (λ, z) = 0 ∈ RM , (λ, z) ∈ R × RN , N, M ∈ N, such that the solutions of the two equations are in one-to-one correspondence. The reduction to finite dimensions in §8.2 is called Lyapunov-Schmidt reduction and leads immediately to a local bifurcation theory based on the implicit function theorem. In particular, it yields a classical relation between a nonlinear problem and its linearization.

1.2 PRINCIPLE OF LINEARIZATION Roughly speaking, the principle of linearization [39] derives from the feeling that when F (λ, 0) = 0 for all λ and solutions with x small are sought, the nonlinear problem F (λ, x) = 0 might as well be replaced with the linear equation ∂x F [(λ, 0)]x = 0, where ∂x F [(λ, 0)] denotes the linearization of F with respect to x at x = 0. Since sin φ = φ + O(|φ|3 ) as φ → 0, the linearization of the elastic-rod problem at (λ0 , 0) is φ

(s) + λ0 φ(s) = 0, s ∈ [0, L], φ
(0) = φ
(L) = 0, λ0 > 0,

6

CHAPTER 1

and this problem has non-trivial solutions if and only if λ0 = (Kπ/L)2 with φ(s) = cos(Kπs/L), K ∈ N. The question is, can any inference be drawn from this about the nonlinear problem (1.3)? The answer is that in quite general situations (including equation (1.3) as a special case) λ0 is a bifurcation point on the line of trivial solutions of (1.2) only if the linearized problem dx F [(λ0 , 0)]x = 0 has a non-trivial solution. The fact that this is also sufficient for bifurcation from the line of trivial solutions for (1.3) (but not in general) is a consequence of the theory of bifurcation from a simple eigenvalue, see §8.4 and §8.5.

1.3 GLOBAL THEORY It is clear from Figure 1.5 that there is more to the solution set of equation (1.3) than is predicted by local theory. Global features of the diagram are not a consequence of finite-dimensional reduction methods alone. We will see in Chapter 9 how local bifurcation theory, the implicit function theorem and some elementary results on real-analytic varieties can be used to piece together a global picture of the solution set of (1.2), without assumptions about the size of the solutions under consideration. Provided some general functional-analytic structure is present and F is real-analytic, the global continuum C of solutions which bifurcates from the trivial solutions at a simple eigenvalue contains a continuous curve R with the following properties. • R = {(Λ(s), κ(s)) : s ∈ [0, ∞)} ⊂ C is either unbounded or forms a closed loop in R × X. • For each s∗ ∈ (0, ∞) there exists ρ∗ : (−1, 1) → R (a re-parameterization) which is continuous, injective, and ρ∗ (0) = s∗ , t → (Λ(ρ∗ (t)), κ(ρ∗ (t)), t ∈ (−1, 1), is analytic. This does not imply that R is locally a smooth curve. (The map σ : (−1, 1) → R2 given by σ(t) = (t2 , t3 ) is real-analytic and its image is a curve with a cusp.) Nor does it preclude the possibility of secondary bifurcation points on R. In particular, since (Λ, κ) : [0, ∞) → R × X is not required to be globally injective; self-intersection of R (as in a figure eight) is not ruled out. • Secondary bifurcation points on the bifurcating branch, if any, are isolated. See Theorem 9.1.1 for a complete statement and §9.3 for an application to the elastic-rod problem. This result about real-analytic global bifurcation from a simple eigenvalue is a sharpened version of a theorem due to Dancer. His general results [24, 26] deal with bifurcation from eigenvalues of higher multiplicity and give the path-connectedness of solutions sets that are not essentially one-dimensional. Since his hypotheses are less restrictive, his conclusions are necessarily somewhat less precise. The topological theory of global bifurcation without analyticity assumptions was developed slightly earlier, first for nonlinear Sturm-Liouville problems

INTRODUCTION

7

(such as (1.3)) by Crandall & Rabinowitz [21], then for partial differential equations by Rabinowitz [50], and for problems with a positivity structure by Dancer [25] and Turner [64]. Their basic tool was an infinite-dimensional topological degree function and the outcome was the existence of a global connected (but not always path-connected) set of solutions. Although it is sometimes possible to argue from the implicit function theorem that the connected set given by topological methods is a smooth curve in R × X, this approach fails if there is a secondary bifurcation point on the bifurcating branch. What is important is that in the analytic case a one-dimensional branch can be followed unambiguously through a secondary bifurcation point. In fact a onedimensional branch is uniquely determined globally by its behaviour in an open set and can be parameterized globally, even when it intersects manifolds of solutions of different dimensions (see §7.5).

1.4 LAYOUT We begin in Chapter 2 with a review, without proofs, of the linear functional analysis needed for nonlinear theory. Chapter 3 introduces the main results from nonlinear analysis, including the inverse and implicit function theorems for functions of limited differentiability in Banach spaces. Chapter 4 covers similar ground for analytic operators and operator equations in Banach spaces. In Chapters 5, 6 and 7 we consider finite-dimensional analyticity with particular regard to analyticity over the field R. We prove the classical theorems of Weierstrass on the reduction of an analytic equation to a canonical form which involves a polynomial equation for one variable in which the coefficients are analytic functions of the other variables. Chapter 8 deals with the finite-dimensional reduction of infinite dimensional problems. When the infinite-dimensional problem involves analytic operators, so does the finite-dimensional reduction and the mapping from solutions of the latter to solutions of the former is also analytic. This chapter is the link between the theory of finite-dimensional analytic varieties and infinite-dimensional problems in Banach spaces. Chapter 9 considers what conclusions can then be drawn about global one-dimensional branches of solutions of real-analytic operator equations. This concludes the abstract theory. Chapter 10 illustrates our discussion of global real-analytic bifurcation theory with a substantial example from mathematical hydrodynamics: the existence question for steady two-dimensional periodic waves on an infinitely deep ocean. There is only one real parameter λ in the problem, the square of the Froude number which represents the speed of the wave. In his 1847 paper [56] Stokes discussed nonlinear waves with small amplitudes using power series. At the time the proof of convergence was very difficult and only in the 1920s did Nekrasov [47] and Levi-Civita [42], independently, settle the question. Nowadays the existence of small-amplitude water waves can be recognised as nothing more complicated than bifurcation from a simple eigenvalue. In an 1880 note, Stokes [57] conjectured the existence of a large amplitude pe-

8

CHAPTER 1

riodic wave with a stagnation point and a corner containing an angle of 120◦ at its highest point. He further speculated that this wave of extreme form marks the limit of steady periodic waves in terms of amplitude (the Stokes wave of greatest height). In Chapter 10 we show how real-analytic global bifurcation theory can account for the existence of waves of all amplitudes from zero up to that of Stokes’ highest wave. See [60] for an account of topological methods applied to the same problem; the conclusions there are, in general, weaker. Almost all the material here is to be found in the literature. The novelty is in the selection and organization of the material with bifurcation theory in mind. Each chapter ends with notes on sources.

Chapter Two Linear Functional Analysis In this chapter we introduce notation and record, without proof, the main results from linear functional analysis used in the sequel.

2.1 PRELIMINARIES AND NOTATION In what follows R denotes the field of real numbers, C denotes the field of complex numbers, N denotes the natural numbers, not including 0; N0 = N ∪ {0}, Z denotes the integers, F denotes R or C in statements which are true for both, Re z, Im z, z and |z| denote the real part, the imaginary part, the complex conjugate and the modulus, respectively, of z ∈ C, Sk denotes the symmetric group of permutations of {1, · · · , k}. We will assume familiarity with concepts such as closedness, completeness, compactness and connectedness in metric spaces, and with linearity, linear independence and dimension in vector spaces over R and C. A maximal connected subset of a metric space M is called a component of M . A subset C of a linear space X over F is said to be convex if for all x1 , x2 ∈ C and every t ∈ [0, 1], (1 − t)x1 + tx2 ∈ C. In other words C contains every line-segment joining any two points of C. A norm on a linear space X over F is an R-valued function · such that

x ≥ 0 for all x ∈ X with equality if and only if x = 0,

αx = |α| x for all x ∈ X and α ∈ F,

x + y ≤ x + y for all x, y ∈ X. Various symbols, for example · , · X or ||| · |||, could be used to denote norms on a space X, but when the intention is clear from the context, we use the generic symbol · in all cases. Two norms ||| · ||| and · on the same linear space X are said to be equivalent if there exist positive constants k, K, such that k x ≤ |||x||| ≤ K x for all x ∈ X.

12

CHAPTER 2

If · is a norm on a linear space X, ρ(x, y) = x − y , x, y ∈ X, defines a metric ρ on X. If the metric space (X, ρ) is complete, (X, · ) is called a Banach space. (A Banach space is said to be real or complex if the field F is R or C, but when the field does not matter it will not be  mentioned.) If a sequence {x } in a Banach space X has n n∈N xn < ∞ then the se  n x of partial sums is Cauchy, and hence convergent, in X. We quence k k=1  ∞  then say that the series is summable in norm, which implies that it is x k=1 k convergent to the same sum irrespective of rearrangement of its terms. L EMMA 2.1.1 Every finite-dimensional linear space with any norm is a Banach space. Any two norms on a finite-dimensional linear space are equivalent. E XAMPLES 2.1.2 (a) The finite-dimensional space FN , N ∈ N, is a Banach space over F with norm | · | where |(z1 , z2 , · · · , zN )| = 2

N 

|zk |2 , (z1 , · · · , zN ) ∈ FN .

k=1

  (b) The linear space C n [0, 1], FN of continuous functions  infinite-dimensional  u = u1 , · · · , uN : [0, 1] → FN , the nth derivatives of which exist on (0, 1) and have continuous extensions to [0, 1], with the norm 

u = sup |u(x)| + |u(n) (x)| : x ∈ (0, 1)}, where  dn u1 d n uN  , u = (u1 , · · · , uN ), , · · · , dxn dxn     is a Banach space. By convention, C [0, 1], FN = C 0 [0, 1], FN . u(n) =

Other Banach spaces, in particular Sobolev spaces of functions, are introduced in [10] and [32]. The following result is due to F. Riesz. L EMMA 2.1.3 A Banach space X is finite-dimensional if and only if the closed unit ball {x ∈ X : x ≤ 1} is compact. Both the closed unit ball and the open unit ball {x ∈ X : x < 1} are convex. The following is a corollary of the open mapping theorem (2.4.2 below). L EMMA 2.1.4 Suppose that · and ||| · ||| are two Banach-space norms on the linear space X and that |||x||| ≤ K x for all x ∈ X. Then ||| · ||| and · are equivalent norms on X.

13

LINEAR FUNCTIONAL ANALYSIS

An inner product ·, · on a linear space X over F is an F-valued function on X × X such that y → x, y is F-linear for each (fixed) x ∈ X,

x, y = y, x for all (x, y) ∈ X × X,

x, x ≥ 0 for all x ∈ X with equality if and only if x = 0. An inner product ·, · is linear with respect to each of the variables separately if F = R, but only with respect to the second variable if F = C. If ·, · is an inner product on X then

x 2 = x, x, x ∈ X, defines a norm on X and the Cauchy-Schwarz inequality, | x, y| ≤ x y for all x, y ∈ X, holds. A complete inner-product space is called a Hilbert space. It is clear that the Banach space FN defined above is a Hilbert space over F because

(z1 , · · · , zN ), (ˆ z1 , · · · , zˆN ) =

N 

zk zˆk

k=1

is an inner-product on FN over F.

2.2 SUBSPACES Suppose that X is a Banach space with norm · and that Y is a linear subspace of X. Then (Y, · ) is a Banach space if and only if Y is closed in X. If X1 , X2 are closed linear subspaces of a Banach space X with the property that X1 ∩ X2 = {0} and, for every x ∈ X, there exist x1 ∈ X1 and x2 ∈ X2 such that x = x1 + x2 , we say that X is the topological direct sum of X1 and X2 and write X = X1 ⊕ X2 . (This notation always has the implication that X1 and X2 are Banach spaces with the norm inherited from X.) The space X2 is called the topological complement of X1 in X, and vice versa. L EMMA 2.2.1 If X = X1 ⊕ X2 , then every x ∈ X can be written in a unique way as x = x1 + x2 , xi ∈ Xi , i = 1, 2. Even when X1 is closed it is not always true that X = X1 ⊕ X2 for some closed linear subspace X2 ; X1 may not have a topological complement. However when X1 is finite-dimensional we have the following. L EMMA 2.2.2 Suppose that X is a Banach space and X1 is a finite-dimensional subspace of X. Then X1 is closed and there exists a closed subspace X2 of X such that X = X1 ⊕ X2 . In this case dim X1 is called the codimension of X2 .

14

CHAPTER 2

2.3 DUAL SPACES Suppose that X is a Banach space over F. A linear functional f on X is a linear mapping from X into F. A linear functional f is said to be bounded if there exists K such that |f (x)| ≤ K x for all x ∈ X. The set of all bounded linear functionals on X is a linear space over F which is usually denoted by X ∗ . It becomes a Banach space when endowed with the norm defined by

f = inf{K ≥ 0 : |f (x)| ≤ K x for all x ∈ X}. The Banach space X ∗ with this norm is called the dual space of X. An important corollary of the Hahn-Banach theorem is the following. L EMMA 2.3.1 Let X be a Banach space. (a) For each x ∈ X there exists fx ∈ X ∗ such that fx (x) = x and fx = 1. (b) Suppose that E is a closed linear subspace of X of codimension 1. Then there exists f ∈ X ∗ such that f (x) = 0 if and only if x ∈ E. In other words, if x0 = 0 and X = span{x0 } ⊕ E there exists f ∈ X ∗ such that f (x0 ) = 1 and ker f = E. Note that in (a) fx need not be unique. For example, let X denote R2 with the norm (x1 , x2 ) = max{|x1 |, |x2 |}. For t ∈ [0, 1] let f(t) (x1 , x2 ) = tx1 + (1 − t)x2 . Then |f(t) (x1 , x2 )| = |tx1 + (1 − t)x2 | ≤ t|x1 | + (1 − t)|x2 | ≤ (x1 , x2 ) , whence f(t) ≤ 1. Since (1, 1) = 1 = f(t) (1, 1) for all t ∈ [0, 1], fx in Lemma 2.3.1 is not unique when x = (1, 1) ∈ X.   E XAMPLES 2.3.2 (a) Let X, ·,  be a Hilbert space. For any x ∈ X with x = 0, the functional defined by fx (y) =

x, y

x

satisfies the requirements of Lemma 2.3.1(a).  (b) Let X denote the Banach space C [0, 1], FN . For u ∈ X \ {0}, choose tu ∈ [0, 1] and θj ∈ [−π, π] such that

u 2 =

N 

u2j (tu )eiθj .

j=1

Now let fu (v) = u −1

N 

uj (tu )vj (tu )eiθj for v ∈ X.

j=1

Then fu ∈ X ∗ , fu = 1 and fu (u) = u , as in Lemma 2.3.1 (a).

LINEAR FUNCTIONAL ANALYSIS

15

It is clear that if X is a Hilbert space then, for any z ∈ X, an element of f ∈ X ∗ can be defined by f (x) = z, x for all x ∈ X. That the converse is also true is the content of the next theorem. T HEOREM 2.3.3 (Riesz Representation Theorem) Let X be a Hilbert space and f ∈ X ∗ . Then there exists a unique xf ∈ X such that f (x) = xf , x and x = xf for all x ∈ X.

2.4 LINEAR OPERATORS Before discussing the boundedness of linear operators between Banach spaces we pause to note that the familiar notion of linearity is intimately connected with the field F. For example, let Cr and Cc denote the complex numbers regarded as a linear space over R, and over C, respectively. (The spaces are the same, only the fields are different.) Now consider the operator L on C defined by Lz = z, the operation of complex conjugation. This is a linear operator on Cr , but not on Cc , an observation which will be important when we consider the Fr´echet differentiability and analyticity of nonlinear operators. Suppose that X, Y are Banach spaces over the same field F. A linear function A : X → Y is said to be a bounded linear operator, written A ∈ L(X, Y ), if there exists K ≥ 0 such that

Ax ≤ K x for all x ∈ X. An important element of L(X, X) for any Banach space X is the identity operator I, defined by Ix = x for all x ∈ X. An obvious element of L(X, Y ) is the zero operator, defined by 0x = 0 ∈ Y for all x ∈ X. It is easily seen that a linear mapping A : X → Y is continuous if and only if it is bounded. It is also clear that when X, Y are Banach spaces over F, L(X, Y ) is a linear space over F with the natural definitions of addition and multiplication by scalars. Even more is true. L EMMA 2.4.1 If X, Y are Banach spaces over F, then L(X, Y ) is a Banach space when endowed with the norm

B = inf{K ≥ 0 : Bx ≤ K x for all x ∈ X}. In particular, L(X, F) = X ∗ . For A ∈ L(X, Y ) let ker(A) = {x ∈ X : Ax = 0 ∈ Y }, range (A) = {y ∈ Y : y = Ax for some x ∈ X}. Both ker(A) ⊂ X and range (A) ⊂ Y are linear spaces and ker(A) is closed because A is continuous. However range (A) need not be closed and for nonlinear analysis it will be important to have hypotheses on A sufficient to guarantee that it is, see §2.7. An operator A ∈ L(X, Y ) is injective if ker(A) = {0}, surjective if

16

CHAPTER 2

range (A) = Y and bijective if it is both. An operator B ∈ L(X, Y ) is said to be a homeomorphism if it is a bijection and B −1 ∈ L(Y, X). T HEOREM 2.4.2 (Open Mapping Theorem) Suppose X and Y are Banach spaces and that B ∈ L(X, Y ) is surjective. Then B(U ) is open in Y for all open sets U ⊂ X. We have already seen an important corollary of the open mapping theorem, Lemma 2.1.4. Here is another. C OROLLARY 2.4.3 (Corollary of Open Mapping Theorem) If X and Y are Banach spaces and B ∈ L(X, Y ) is a bijection, then B is a homeomorphism. This result will be important when we have to check the hypotheses of the implicit or inverse function theorems which require a certain operator to be a homeomorphism between Banach spaces. It shows that being bijective and bounded is enough. For A ∈ L(X, X) let Ak = A ◦ A ◦ · · · ◦ A . k times

2.5 NEUMANN SERIES Let A ∈ L(X, X) be such that A < 1. Then the sequence of partial sums N k k=0 A is a Cauchy sequence ∞ which converges in the Banach space L(X, X) and the limit is denoted by k=0 Ak . It is easily seen that ∞ ∞ ∞

 

   (I − A) Ak = I = Ak (I − A), (I − A)−1 = Ak , k=0

k=0

(2.1)

k=0

and I − A is a homeomorphism. The series on the right is called the Neumann series of A. Note that because ∞ ∞     k ≤ A

A k < ∞,   k=0

k=0

the series in (2.1) is summable in norm. Now suppose that T ∈ L(X, Y ) is a homeomorphism. If S ∈ L(X, Y ) is such that T −1 (S − T ) < 1, it follows that I + T −1 (S − T ) : X → X has a bounded inverse given by a Neumann series. Since S = T (I + T −1 (S − T )), the existence of S −1 ∈ L(Y, X), and hence the following result, is now immediate. L EMMA 2.5.1 If T ∈ L(X, Y ) is a homeomorphism and S ∈ L(X, Y ) has

S − T < T −1 −1 , then S is a homeomorphism. In fact the homeomorphisms form an open set in L(X, Y ) on which the mapping S → S −1 is continuous from L(X, Y ) to L(Y, X). If A ∈ L(X, Y ) then A∗ ∈ L(Y ∗ , X ∗ ) is the linear operator, called the conjugate of A, defined by (A∗ f )(x) = f (A(x)) for all x ∈ X and f ∈ Y ∗ .

17

LINEAR FUNCTIONAL ANALYSIS

2.6 PROJECTIONS AND SUBSPACES Suppose that X is a Banach space and that P ∈ L(X, X) has the property that P (P x) = P x for all x ∈ X. Then P is said to be a projection or a projection operator. (In particular, projection operators are bounded.) Note that P is a projection if and only if I − P is a projection. Clearly the zero operator and the identity on X are projections. The connection between projection operators and topological direct sums is summarized in the following two propositions. P ROPOSITION 2.6.1 is a projection. Then

Suppose that X is a Banach space and that P : X → X

ker(P ) and range (P ) are closed subspaces of X; range (P ) = ker(I − P ) and ker(I − P ) = range (P ); ker(P ) ⊕ range (P ) = X. P ROPOSITION 2.6.2 Suppose that X is a Banach space with X = X1 ⊕ X2 . By Lemma 2.2.1, x ∈ X can be written in a unique way as x = x1 + x2 where x1 ∈ X1 and x2 ∈ X2 . Let P x = x1 , so that (I − P )x = x2 . Then both P and I − P are projections on X. P is called the projection onto X1 parallel to X2 . That P and I − P defined in the second proposition are linear follows easily from the definition of a topological direct sum. What is not so obvious is that P , so defined, is a bounded operator. This is yet another corollary of the open mapping theorem 2.4.2. Boundedness of projections is essential for our purposes. Note that a closed subspace X1 of X alone does not define a projection and there is no such notion as “the projection onto X1 ”. Indeed, if X1 has no topological complement, then there does not exist a projection from X onto X1 . However, because of Lemma 2.2.2 we have the following. L EMMA 2.6.3 If X1 is a finite-dimensional subspace of a Banach space X, then there exists a projection P on X with X1 = range (P ). D EFINITION 2.6.4 Suppose X1 , · · · , Xn are Banach spaces over a field F. Then the product space of n-tuples (x1 , · · · , xn ), xi ∈ Xi , is a Banach space denoted by X1 × · · · × Xn . Many equivalent norms may be defined on X1 × · · · × Xn , but we shall assume that   n 

xk 2 .

(x1 , · · · , xn ) =  k=1

When each of the spaces Xk is a Hilbert space, the product space is also a Hilbert space with inner product 

n   (x1 , · · · , xn ), (z1 , · · · zn ) =

xk , zk , k=1

18

CHAPTER 2

for (x1 , · · · , xn ), (z1 , · · · , zn ) ∈ X1 × · · · × Xn . From time to time it will be convenient to identify the space Xk with {0} × · · · × {0} ×Xk × {0} × · · · × {0} ⊂ X1 × · · · × Xn ,



 k−1 terms

n−k terms

and hence with a closed subspace of the product space. There is an obvious projection of the product space onto the space identified with Xk , parallel to the product of the other spaces.

2.7 COMPACT AND FREDHOLM OPERATORS Suppose that (M, ρ) is a compact metric space. Let C(M, F) denote the linear space of continuous functions u : M → F with the norm

u = max{|u(x)| : x ∈ M }. Then C(M, F) is a Banach space. The following characterizes its compact sets. T HEOREM 2.7.1 (Ascoli-Arzel`a Theorem) Suppose that M is a compact metric space. A set B ⊂ C(M, F) has compact closure if and only if (i) there exists k such that u ≤ k for all u ∈ B, and (ii) given > 0 there exists δ > 0 (independent of u) such that for all u ∈ B |u(x) − u(y)| < if ρ(x, y) < δ. Let X and Y be Banach spaces. An operator K ∈ L(X, Y ) is said to be compact if every bounded sequence {xn } ⊂ X has a subsequence {xnk } for which {K(xnk )} converges in Y . In finite dimensions, all linear operators are bounded, and therefore compact, by Lemma 2.1.3. As a consequence, if Y is finite dimensional and X is a Banach space, then any linear operator L : Y → X is compact and any A ∈ L(X, Y ) is compact. Note that if W, X, Y and Z are Banach spaces and K ∈ L(X, Y ) is compact, B ∈ L(Z, X) and C ∈ L(Y, W ), then C ◦ K ◦ B ∈ L(Z, W ) is compact. Theorem 2.7.1 leads to an important class of compact operators. D EFINITION 2.7.2 Suppose that X and Y are linear spaces over F and X ⊂ Y . Now suppose that (X, | · |) and (Y, · ) are Banach spaces for which the mapping ι : X → Y , given by ι(x) = x ∈ Y for all x ∈ X, is bounded. We say that the embedding of X in Y is continuous and ι is called the embedding operator. If ι is compact, we say that X is compactly embedded in Y . example of compact embeddings is the folE XAMPLE 2.7.3 The  prototypical    lowing. Let X = C 1 [0, 1], F , Y = C [0, 1], F , defined in Example 2.1.2 (b), and let Kf = f ∈ Y for f ∈ X. Then K is compact, by the Ascoli-Arzel`a theorem and the mean-value theorem for functions of one variable. In this example K coincides with the embedding operator ι from X ⊂ Y into Y . Another source of compact operators is the following.

LINEAR FUNCTIONAL ANALYSIS

19

L EMMA 2.7.4 When X and Y are Banach spaces the compact operators form a closed linear subspace of L(X, Y ). In particular, if there exists a sequence {An } ⊂ L(X, Y ) such that An − A → 0 as n → ∞ and An has finite-dimensional range for each n, then A is a compact operator. An operator A ∈ L(X, Y ) is a Fredholm operator of index p if ker(A) has finite dimension n; range (A) is closed and has finite codimension r; p = n − r. Clearly any homeomorphism from X to Y is a Fredholm operator of index zero. The following result for compact operators in Banach spaces contains the dimension theorem in linear algebra as a special case and explains the central rˆole of compact operators in functional analysis. T HEOREM 2.7.5 (Fredholm Alternative) Let K ∈ L(X, X) be compact. Then I − K ∈ L(X, X) and I − K ∗ ∈ L(X ∗ , X ∗ ) are Fredholm operators of index zero. Moreover dim ker(I − K) = dim ker(I − K ∗ ) = codim range (I − K) = codim range (I − K ∗ ). The following criterion ensures that an operator B ∈ L(X, Y ), for different spaces X, Y , is Fredholm in an infinite dimensional setting. T HEOREM 2.7.6 Suppose X and Y are Banach spaces, K ∈ L(X, Y ) is compact and T ∈ L(X, Y ) is a homeomorphism. Then B = T + K is Fredholm with index zero. Proof. It suffices to notice that B = T + K = T (I + T −1 K) and, since T, T −1 are bounded, B ∈ L(X, Y ) is Fredholm with index zero if and only if I + T −1 K ∈ L(X, X) is Fredholm with index zero. However T −1 K ∈ L(X, X) is compact, and the result follows from Theorem 2.7.5. P ROPOSITION 2.7.7 [67] Let X, Y be Banach spaces. Then the set of Fredholm operators is an open set in L(X, Y ) and the Fredholm index of operators is constant on the components of this set. D EFINITION 2.7.8 Let ι ∈ L(X, Y ) denote the continuous embedding of a Banach space X in a Banach space Y . Suppose that λ0 ∈ F and A ∈ L(X, Y ) are such that λ0 ι − A is Fredholm of index  zero, that ker(λ  0 ι − A) is one-dimensional over F, and that range (λ0 ι − A) ∩ ι ker(λ0 ι − A) = {0}. Then we say that λ0 is a simple eigenvalue of A. (When X = Y is finite-dimensional, this is equivalent to saying that λ0 is an eigenvalue of A of algebraic multiplicity 1.) An element ξ0 ∈ X \ {0} with Aξ0 = λ0 ι ξ0 is called an eigenvector of A corresponding to the eigenvalue λ0 .

20

CHAPTER 2

L EMMA 2.7.9 Let

Suppose that λ0 is a simple eigenvalue of A with eigenvector ξ0 .

  Y1 = range (λ0 ι − A) and X1 = ι−1 (Y1 ) = X ∩ Y1 . Then X = X1 ⊕ span {ξ0 } and λ0 ι − A is a homeomorphism from X1 to Y1 with the norms inherited from X and Y . Proof. For any y ∈ Y , y = αιξ0 + y1 , y1 ∈ Y1 , α ∈ F, since Y1 has codimension / Y1 . In particular, for x ∈ X, ιx = αιξ0 + y1 and x − αξ0 ∈ ι−1 (Y1 ) = 1 and ιξ0 ∈ X1 . Hence x = x1 + αιξ0 for some x1 ∈ X1 and α ∈ F. Next, note that X1 = ι−1 (Y1 ) is closed in X since  Y1 is closed in Y and ι : X → Y is continuous. Since X1 ∩ span {ξ0 } = ι−1 Y1 ∩ span {ι ξ0 } = {0}, it follows that X = X1 ⊕ span {ξ0 }. So λ0 ι − A ∈ L(X1 , Y1 ) is a bijection, and hence a homeomorphism. This completes the proof.

2.8 NOTES ON SOURCES This is standard material, to be found, for example, in the books of Brezis [10], Friedman [32], Kreyszig [41], Rudin [51] or Taylor [58]. The theory of Fredholm operators and their indices is covered in the books by Kato [35] and Wloka [67].

Chapter Three Calculus in Banach Spaces We turn to our main topic, nonlinear operators between Banach spaces. “Big O” and “little o” notation. Suppose that f and g are functions defined from a neighbourhood of a in a Banach space X to a Banach space Y . We write f (x) = g(x) + o(x − a) as x → a if lim

x→a

f (x) − g(x)

= 0,

x − a

and f (x) = g(x) + O(x − a) as x → a if lim sup x→a

f (x) − g(x)

< ∞.

x − a

´ 3.1 FRECHET DIFFERENTIATION Suppose that X and Y are Banach spaces, U ⊂ X is open, and F : U → Y . D EFINITION 3.1.1 The function F is Fr´echet differentiable at x0 ∈ U if there exists A ∈ L(X, Y ) such that

F (x0 + h) − F (x0 ) − Ah

= 0.

h

0 0 and choose δ > 0 such that

∂y F [(x, y)] − ∂y F [(x0 , y0 )] ≤ if (x − x0 , y − y0 ) ≤ δ. For any x, y with (x, y) < δ let u(t) = F (x0 + x, y0 + ty) − t∂y F [(x0 , y0 )]y. Then, by the Chain Rule,  

du[t] = ∂y F [(x0 + x, y0 + ty)] − ∂y F [(x0 , y0 )] y

≤ ∂y F [(x0 + x, y0 + ty)] − ∂y F [(x0 , y0 )] y

≤

y since (x, y) < δ. Hence, by Lemma 3.1.4, I1 = u(1) − u(0) ≤

y ≤ ( x + y ) if (x, y) < δ. Hence I1 = o( (x, y) ) as (x, y) → 0. Combining these estimates for I1 and I2 yields that dF [(x0 , y0 )] exists and dF [(x0 , y0 )](x, y) = ∂x F [(x0 , y0 )]x + ∂y F [(x0 , y0 )]y. This completes the proof.

25

CALCULUS IN BANACH SPACES

Suppose that F is Fr´echet differentiable on U and x → dF [x] is continuous from U ⊂ X into the Banach space L(X, Y ). Then we say that F is continuously Fr´echet differentiable on U , or that F is of class C 1 on U . This is written F ∈ C 1 (U, Y ). As a consequence of the preceding results, a function defined on a product of Banach spaces is continuously differentiable if and only if each of its partial derivatives is continuously differentiable. Here are three examples where C 1 functions arise; the first will be familiar and the third shows that nothing can be taken for granted. E XAMPLE 3.1.8 (Finite Dimensions) An important special case of this observation occurs when f : RN → RM is given by   f (x1 , · · · , xN ) = f 1 (x1 , · · · , xN ), · · · , f M (x1 , · · · , xN ) . Then f is continuously Fr´echet differentiable if and only if each ∂xj f i , 1 ≤ i ≤ M , 1 ≤ j ≤ N , is a continuous R-valued function of (x1 , · · · , xN ). When U ⊂ FN is open and f : FN → FM , the Fr´echet derivative of f at a point is sometimes called the total derivative to distinguish it from partial derivatives. From advanced calculus courses we may recall that the existence of a total derivative implies the existence of all the partial derivatives of the component functions of f at the same point, but not vice versa. However if all the partial derivatives exist and are continuous at every point of U , then the total derivative also exists and f ∈ C 1 (U, FM ). We have now observed that this theory remains valid in infinite dimensions with no extra difficulties. E XAMPLE 3.1.9 (Nemytskii Operators on C) In Examples 2.1.2 (b) we de  N M fined the Banach space C [0, 1], FN . Suppose that  f : F N → F is continuously  differentiable. Then a Nemytskii operator F : C [0, 1], F → C [0, 1], FM can be defined by composition:   F (u)(t) = f (u(t)), t ∈ [0, 1], u ∈ C [0, 1], FN , and it is not difficult to see that the nonlinear operator F is of class C 1 in this setting. E XAMPLE 3.1.10 (Nemytskii Operators on Lp ) Let Lp [0, 1], 1 ≤ p < ∞, denote the Banach space of pth power Lebesgue integrable real-valued ‘functions’ u : [0, 1] → F with the norm


1 1/p

u p = |u(s)|p ds . 0

Suppose that g : R → R, g(0) = 0 and let G(u)(x) = g(u(x)), x ∈ [0, 1], u ∈ Lp [0, 1]. Now suppose that G maps Lp [0, 1] into Lp [0, 1] and is Fr´echet differentiable at 0 ∈ Lp [0, 1]. Then g(t) = bt for some b ∈ R. To see this, suppose that g(s)/s = g(t)/t

26

CHAPTER 3

for some non-zero s, t ∈ R. For δ ∈ [0, 1), let uδ = tχ[0,δ] and vδ = sχ[0,δ] , where χ[a,b] is the function with value 1 on [a, b] and zero otherwise. Then vδ = suδ /t and clearly uδ p and vδ p tend to 0 as δ → 0. Now suppose that dG[0] = L ∈ L(Lp [0, 1], Lp [0, 1]) exists. Then, since G(0) = 0, G(uδ ) − Luδ p / uδ p → 0 and

(t/s)G(vδ ) − Luδ p

G(vδ ) − Lvδ p = →0

uδ p

vδ p as δ → 0. Thus |(t/s)g(s) − g(t)|

(t/s)G(vδ ) − G(uδ ) p → 0 as δ → 0. = |t|

uδ p This contradiction shows that, in the setting of pth power Lebesgue integrable functions, Fr´echet differentiability at 0 of a Nemytskii operator implies that the operator in question is affine (linear + constant). The notion of a compact linear operator in §2.7 has a natural analogue for nonlinear operators, although a certain amount of care should be exercised (see the remark following the definition). D EFINITION 3.1.11 A (nonlinear) function F from a subset U of a Banach space X into a Banach space Y is said to be compact if F (W ) is compact in Y when W ⊂ U is bounded in X. A compact linear operator from X to Y maps bounded sets to bounded sets and is therefore continuous. If F is nonlinear and compact, it need not be continuous. A continuous compact operator is sometimes called completely continuous. Many problems can be written as equations which involve nonlinear compact operators in Banach spaces. It is therefore useful to note that the linearizations of such equations involve compact linear operators. L EMMA 3.1.12 (Differentiation of Compact Operators) Suppose U is open in X, F : U → Y is compact, and dF [x0 ] exists at x0 ∈ U . Then dF [x0 ] ∈ L(X, Y ) is compact. Proof. Let {xn } ⊂ X be bounded. Then the compactness of F and a diagonalization argument means that there is no loss of generality in supposing that {x0 + (xn /k)} ⊂ U and {F (x0 + (xn /k))}n∈N is convergent as n → ∞ for each fixed k ∈ N with k ≥ K sufficiently large. We want to show that {dF [x0 ](xn )}n∈N is Cauchy in Y . Let M = sup{ xn : n ∈ N} and let > 0 be given. Then there exists δ > 0 such that

R(h) = F (x0 + h) − F (x0 ) − dF [x0 ]h ≤ For all n, m, k ∈ N,

h

if h < δ. 2M

  dF [x0 ]xn − dF [x0 ]xm = k dF [x0 ](xn /k) − dF [x0 ](xm /k)  x   xn    xn  xm  m =k R −R + F x0 + − F x0 + . k k k k

CALCULUS IN BANACH SPACES

27

Let kˆ (fixed) be such that kˆ ≥ K and kˆ > M/δ. Then

dF [x0 ]xn − dF [x0 ]xm

   xm xn  xm   xn ≤ kˆ R( ) + R( ) + F x0 + − F x0 +

ˆ kˆ kˆ kˆ   k xn xm ≤ kˆ { xn + xm } + F (x0 + ) − F (x0 + )

ˆ ˆ 2M k k kˆ ˆ − F (x0 + (xm /k)) . ˆ ˆ (x0 + (xn /k)) ≤ + k F ˆ n∈N is Cauchy, the sequence {dF [x0 ]xn }n∈N is also Since {F (x0 + (xn /k))} Cauchy, and the result follows. The converse is false: the derivative of a function at a point may be compact without 2 the operator itself being  compact. For example, let F (u) = u for u ∈ C[0, 1]. Then dF [0] = 0 ∈ L C[0, 1], C[0, 1] , a compact operator. Now let un (x) = 0 for x ∈ [n−1 , 1] and un (x) = 1 − nx for x ∈ [0, n−1 ]. Clearly {F (un )} is not relatively compact in C[0, 1] and hence F is not a compact operator. In finite-dimensional spaces, all continuous functions on closed sets are compact. While the compactness of F implies the compactness of dF [x] ∈ L(X, Y ), it does not imply the compactness of dF : X → L(X, Y ), even in one dimension. Here is an example. Let f (0) = 0 and f (x) = x2 sin(1/x2 ) for x = 0. Since df is everywhere defined with df [0] = 0, but df is not bounded in a neighbourhood of 0, df : R → L(R, R) is not compact.

3.2 HIGHER DERIVATIVES Suppose that F is continuously Fr´echet differentiable on U . If dF : U → L(X, Y ) is itself differentiable at x0 ∈ U , we say that the second Fr´echet derivative of F at x0 ∈ U exists. Note that   d dF [x0 ] ∈ L(X, L(X, Y )), d(dF )[x0 ]x1 ∈ L(X, Y ), d(dF )[x0 ](x1 )(x2 ) ∈ Y, for all x1 , x2 ∈ X, and the mappings d(dF )[x0 ](·)(x2 ) and x2 → d(dF )[x0 ](x1 )(·) are elements of L(X, Y ). T HEOREM 3.2.1 Suppose that F : U → Y and the second Fr´echet derivative of F exists at x0 ∈ U (open in X). Then d(dF )[x0 ](x1 )(x2 ) = d(dF )[x0 ](x2 )(x1 ), (x1 , x2 ) ∈ X 2 . Proof. For x1 , x2 sufficiently close to 0 in X let Φ(x1 , x2 ) = F (x0 + x1 + x2 ) − F (x0 + x1 ) − F (x0 + x2 ) + F (x0 ).

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CHAPTER 3

Clearly Φ(x1 , x2 ) = Φ(x2 , x1 ). To prove the required result it suffices to show that

Φ(x2 , x1 ) − d(dF )[x0 ](x1 )(x2 ) = o( x1 2 + x2 2 ) as x1 + x2 → 0, because, for any t ∈ R and x1 , x2 ∈ X (fixed) it then follows that t2 d(dF )[x0 ](x1 )(x2 ) − d(dF )[x0 ](x2 )(x1 )

= d(dF )[x0 ](tx1 )(tx2 ) − d(dF )[x0 ](tx2 )(tx1 )

  = Φ(tx1 , tx2 ) − d(dF )[x0 ](tx2 )(tx1 )   − Φ(tx2 , tx1 ) − d(dF )[x0 ](tx1 )(tx2 )

= o( tx1 2 + tx2 2 ) = o(t2 ) as t → 0. Dividing through by t2 and letting t → 0 yields the required result. Now

Φ(x2 , x1 ) − d(dF )[x0 ](x1 )(x2 )

≤ Φ(x2 , x1 ) − dF [x0 + x1 ]x2 + dF [x0 ]x2

+ dF [x0 + x1 ]x2 − dF [x0 ]x2 − d(dF )[x0 ](x1 )(x2 )

= I1 (x1 , x2 ) + I2 (x1 , x2 ), say, and   I2 (x1 , x2 ) ≤ dF [x0 + x1 ] − dF [x0 ] − d(dF )[x0 ]x1  x2

= x2 (o( x1 )) as x1 → 0, since d(dF )[x0 ] is the derivative of dF : X → L(X, Y ) at x0 ∈ X. Hence for

> 0 there exists δ > 0 such that I2 (x1 , x2 ) ≤

x1

x2 if x1 ≤ δ, whence I2 (x1 , x2 ) ≤ 12 ( x1 2 + x2 2 ) if x1 + x2 ≤ δ. This shows that I2 (x1 , x2 ) = o( x1 2 + x2 2 ) as x1 + x2 → 0. For t ∈ [0, 1] let u(t) = F (x0 + x1 + tx2 ) − F (x0 + tx2 ) − t(dF [x0 + x1 ]x2 − dF [x0 ]x2 ). Then I1 (x1 , x2 ) = u(1) − u(0) ≤ sup{ du[t] : 0 ≤ t ≤ 1},

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CALCULUS IN BANACH SPACES

by Lemma 3.1.4. Now du[t] is bounded by  

x2 dF [x0 + x1 + tx2 ] − dF [x0 + tx2 ] − dF [x0 + x1 ] + dF [x0 ]  = x2 {dF [x0 + x1 + tx2 ] − dF [x0 ] − d(dF )[x0 ](x1 + tx2 )} − {dF [x0 + tx2 ] − dF [x0 ] − d(dF )[x0 ](tx2 )}  − {dF [x0 + x1 ] − dF [x0 ] − d(dF )[x0 ]x1 } = x2 {o( x1 + tx2 ) + o( tx2 ) + o( x1 )} as x1 + x2 → 0. Hence given > 0 there exists δ > 0 such that if x1 +

x2 ≤ δ, then I1 (x1 , x2 ) ≤

x2 ( x1 + tx2 + tx2 + x1 ) ≤ 2

x2 ( x1 + x2 ) ≤ 3 ( x1 2 + x2 2 ). Therefore I1 (x1 , x2 ) = o( x1 2 + x2 2 ) as x1 + x2 → 0. Combining the estimates for I1 and I2 yields the estimate for Φ(x2 , x1 ) − d(dF )[x0 ](x1 )(x2 )

needed to complete the proof. If the second Fr´echet derivative exists at every point of U we say that F is twice differentiable on U , and if d(dF ) is continuous from U to L(X, L(X, Y )) we say that F is twice continuously differentiable on U . We then write F ∈ C 2 (U, Y ) and say that F is of class C 2 on U . It is usual to write d(dF )[x0 ](x1 )(x2 ) as d2 F [x0 ](x1 , x2 ). Remember that the order of x1 and x2 does not matter. D EFINITION 3.2.2 An operator b : X × X → Y with the property that b(x, ·) and b(·, x) are linear on X for each fixed x ∈ X is said to be F-bilinear. If in addition, b(x1 , x2 ) = b(x2 , x1 ) for all (x1 , x2 ) ∈ X × X, b is called a symmetric F-bilinear operator from X into Y . When Y = F, b is called a bilinear form. When the field F is given by the context, we can omit the prefix F and refer to bilinear forms and operators. When d2 F [x] exists it is a symmetric bilinear operator from X × X to Y . It is clear that the nth Fr´echet derivative of F can be defined by induction. For x0 ∈ U , x1 , · · · , xn ∈ X and 1 ≤ k ≤ n, let dk F [x0 ](x1 , · · · , xk ) be defined recursively as follows. d1 F [x0 ](x1 ) = dF [x0 ]x1 , d2 F [x0 ](x1 , x2 ) = d(dF )[x0 ](x1 )(x2 ),   dk F [x0 ](x1 , · · · , xk ) = d · · · d(dF ) · · · [x0 ](x1 )(x2 ) · · · (xk ).

 k−1 parentheses

P ROPOSITION 3.2.3 Suppose F maps a neighbourhood U of x0 in X into Y and that the nth Fr´echet derivative of F at x0 exists. Then x → dn F [x0 ](x1 , · · · , xk−1 , x, xk+1 , xn ), x ∈ X,

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CHAPTER 3

is linear for each k ∈ {1, · · · , n}, and dn F [x0 ](x1 , · · · , xn ) = dn F [x0 ](xπ(1) , · · · , xπ(n) ) for all π ∈ Sn (the symmetric group). Proof. By definition, the nth derivative of F at x0 , dn F [x0 ] ∈ L(X, L(X, L(X, · · · , L(X, Y ) · · · ), (L repeated n times) is identified with dn F [x0 ](x1 , x2 , · · · , xn ). It is clear from the definition that dn F [x0 ] is linear in each of the variables (x1 , · · · , xn ) separately. We need only show its symmetry. Theorem 3.2.1 shows the result for n = 2. Suppose n ≥ 3 and (x3 , · · · , xn+1 ) ∈ X n−1 and define g : U → Y by g(x) = dn−1 F [x](x3 , · · · , xn+1 ). By Theorem 3.2.1, for any (x1 , x2 ) ∈ X × X, dn+1 F [x0 ](x1 , x2 , · · · , xn+1 ) = d2 g[x0 ](x1 , x2 ) = d2 g[x0 ](x2 , x1 ) = dn+1 F [x0 ](x2 , x1 , · · · , xn+1 ). (3.1) We proceed by induction. Suppose that dn+1 F [x0 ] exists and that for all x ∈ U and (x1 , · · · , xn ) ∈ X n , dn F [x](x1 , · · · , xn ) = dn F [x](xπ(1) , · · · , xπ(n) ), where π ∈ Sn , the symmetric group. A differentiation with respect to x at x0 now gives, for x ∈ X, dn+1 F [x0 ](x, x1 , · · · , xn ) = dn+1 F [x0 ](x, xπ(1) , · · · , xπ(n) ).

(3.2)

n+1

F [x0 ] is symmetric with respect to interFrom (3.1) and (3.2) it follows that d changing any pair of its components. This completes the proof. R EMARK 3.2.4 (Higher Mixed Derivatives) Suppose that the second Fr´echet derivative of F : U ⊂ X × Y → Z exists at (x0 , y0 ). Then ∂x F : U → L(X, Z) is well defined and has a partial derivative with respect to y     ∂y ∂x F [(x0 , y0 )] ∈ L Y, L(X, Z) . A similar statement holds when x and y are interchanged. It is clear from the definitions that for x ∈ X, y ∈ Y , 

    ∂y ∂x F [(x0 , y0 )](y) (x) = d2 F [(x0 , y0 )] (x, 0), (0, y)      = d2 F [(x0 , y0 )] (0, y), (x, 0) = ∂x ∂y F [(x0 , y0 )](x) (y).     When F has a second Fr´echet derivative, ∂y ∂x F = ∂x ∂y F in this sense and 2 2 we denote it by ∂x,y F [(x0 , y0 )] = ∂y,x F [(x0 , y0 )]. The second partial derivative 2 with respect to x is denoted by ∂x2 F [(x0 , y0 )].

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CALCULUS IN BANACH SPACES

3.3 TAYLOR’S THEOREM Here is the extension to a Banach space setting of the classical theorem on the difference between a function and its nth order Taylor polynomial, familiar from real-variable theory. For x ∈ X and k ∈ N0 define dk F [x0 ]xk = dk F [x0 ] (x, · · · , x), x0 ∈ U,  k times

with the convention that d0 F [x0 ]x0 = F (x0 ), x ∈ X, x0 ∈ U. T HEOREM 3.3.1 (Taylor’s Theorem) Suppose X, Y Banach spaces, U ⊂ X open and convex, and F ∈ C n+1 (U, Y ), n ∈ N0 . Let x, x0 ∈ U . Then n  1 k d F [x0 ](x − x0 )k = Rn (x, x0 ) F (x) − k! k=0

where

Rn (x, x0 ) ≤

x − x0 n+1 sup dn+1 F [(1 − t)x0 + tx] . (n + 1)! 0≤t≤1

Proof. Suppose that f : [0, 1] → R has n + 1 derivatives which are continuous on [0, 1]. Then it is elementary to prove, using induction and the fundamental theorem of calculus, that, for t ∈ [0, 1], f (t) −

n  f (k) (0)tk k=0

k!
t


x1




x2

= 0

0

0


···

xn

f (n+1) (s)dsdxn · · · dx2 dx1 ,

0

and therefore n   f (k) (0)tk  tn+1  sup{|f n+1 (s)| : s ∈ [0, 1]}. f (t) − ≤ k! (n + 1)! k=0

Now for x, x0 in the statement let y ∗ ∈ Y ∗ be such that y ∗ = 1 and n

  1 k d F [x0 ](x − x0 )k y ∗ F (x) − k! k=0

n    1 k   = F (x) − d F [x0 ](x − x0 )k , k! k=0

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CHAPTER 3

and, for t ∈ [0, 1], let     f (t) = Re y ∗ F (tx + (1 − t)x0 . Then in the statement of the theorem, n

  1 k

Rn (x − x0 ) = y ∗ F (x) − d F [x0 ](x − x0 )k k! k=0

=f (1) −

n  f (k) (0)1k

k!

k=0

≤ because

≤

x − x0

(n + 1)!

n+1

1 sup{|f n+1 (s)| : s ∈ [0, 1]} (n + 1)!

sup dn+1 F [(1 − t)x0 + tx] , 0≤t≤1

  f (n+1) (t) = Re y ∗ dn+1 F [(1 − t)x + tx0 ](x − x0 )n+1 .

This completes the proof. D EFINITION 3.3.2 Suppose that X and Y are Banach spaces and U ⊂ X is open. If F : U → Y has Fr´echet derivatives of all orders up to n at x0 ∈ U , then the nth order Taylor polynomial of F at x0 is Tn [F ]x0 (x) =

n  1 k d F [x0 ](x − x0 )k . k!

k=0

When it is defined the infinite series ∞  1 k d F [x0 ](x − x0 )k , k!

k=0

whether it converges or not, is called the Taylor series of F at x0 . When it does converge, there is no a priori reason for its limit to be F (x); in general it is not.

3.4 GRADIENT OPERATORS Before the general case, we consider the real Banach space Rn where it is common to speak of gradient vectors and Jacobian matrices in the context of differentiation. Let ·, · denote an inner product on Rn . If g : Rn → R is Fr´echet differentiable at x0 its derivative is an element of L(Rn , R) = Rn ∗ . Hence, by the Riesz representation theorem, there exists a unique y0 in Rn such that dg[x0 ]x = y0 , x for all x ∈ Rn .

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CALCULUS IN BANACH SPACES

We say that y0 ∈ Rn is the gradient of g at x0 with respect to the inner product

·, ·. However there are many inner products on Rn . For example, if {e1 , · · · , en } is a basis for Rn then

x, y =

n 

αi βi where x =

i=1

n 

αi ei and y =

i=1

n 

βi ei

i=1

defines a different inner product for each choice of basis. Clearly the gradient depends on the choice of inner product. With the above inner product, n n  

 αi ei = αi dg[x0 ]ei

y0 , x = dg[x0 ]x = dg[x0 ]

=

n 

i=1

αi ei ,

i=1

i=1

n 

 (dg[x0 ]ei )ei ,

i=1

and hence the gradient of g at x0 is given by n 

(dg[x0 ]ei )ei ∈ Rn .

i=1

If the standard basis and standard inner product on Rn have been chosen, then the directional derivative dg[x0 ]ei is given by the classical partial derivative ∂g/∂xi x 0 and so the gradient of g is given by the familiar formula  ∇g(x0 ) = (∂g/∂x1 , · · · , ∂g/∂xn )x , 0 where it is understood that g(x) is given as a function of the components of x with respect to the standard basis. Now we consider the second derivative d2 g[x0 ]. This is a symmetric bilinear form on Rn . Therefore, for each x ∈ Rn , the Riesz representation theorem implies the existence of a unique point yx such that d2 g[x0 ](x, y) = yx , y, for all y ∈ Rn and yx depends linearly on x. Hence there exists a linear transformation L on Rn such that Lx = yx . Since d2 g[x0 ] is symmetric and F = R in this example,

Lx, y = d2 g[x0 ](x, y) = d2 g[x0 ](y, x) = Ly, x = x, Ly. Hence L, the Jacobian transformation of g at x0 , is a symmetric operator. The matrix which represents L with respect to the basis used in the definition of the inner product is called the Jacobian matrix. As before, when the standard basis and inner product are chosen, we find a familiar formula for the components of the Jacobian matrix  Li,j = (∂ 2 g(x0 )/∂xi ∂xj )x , i, j = 1, · · · , n. 0

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CHAPTER 3

Now suppose that X is a Hilbert space over F with inner product ·, · and suppose that U ⊂ X is open. If g : U → F is Fr´echet differentiable at x0 ∈ U then dg[x0 ] ∈ X ∗ and, by the Riesz representation theorem, there exists ∇g(x0 ) ∈ X such that dg[x0 ]y = ∇g(x0 ), y for all y ∈ X. The element ∇g(x0 ) is called the gradient of g at x0 (where the rˆole of X is understood). If G : U → X coincides with ∇g on U we say that G is a gradient operator. More generally we have the following definition. D EFINITION 3.4.1 Suppose that Y is a Banach space over F which is continuously embedded in a Hilbert space (X, ·, ·), let U ⊂ Y be open and let g : U → F be Fr´echet differentiable at y0 ∈ U . Then dg[y0 ] ∈ L(Y, F) = Y ∗ . Suppose that there exists xg ∈ X such that dg[y0 ]y = xg , y, for all y ∈ Y, then we say that xg is the gradient in X of g at y0 and write xg = ∇X g(y0 ). Therefore, when it exists, dg[y0 ]y = ∇X g(y0 ), y for all y ∈ Y. (When X is clear from the context, we use ∇ instead of ∇X for the gradient.) Now suppose  that g : U → F has a gradient in X, and that ∇X g : U → X has a derivative d ∇X g [y0 ], which we denote by D2 g[y0 ] ∈ L(Y, X), at every point y0 ∈ U . Since, for y ∈ Y , D2 g[y0 ]y ∈ X and Y is continuously embedded in X, we can define y0 y ∈ Y ∗ by y0 y(z) = D2 g[y0 ]y, z for all z ∈ Y where |y0 y(z)| ≤ D2 g[y0 ] L(Y,X) y Y z Y for all z ∈ Y. Since Y is continuously embedded in X, for h ∈ Y such that y0 + h ∈ U ,

dg[y + h] − dg[y ] −  h  0 0 y0 z

h

 ∇X g(y0 + h) − ∇X g(y0 ) − D2 g[y0 ]h  = , z → 0 as h Y → 0,

h

uniformly for z ∈ Y with z Y ≤ 1. Therefore d2 g[y0 ](y, x) = y0 y(x) = D2 g[y0 ]y, x.

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CALCULUS IN BANACH SPACES

D EFINITION 3.4.2

If G : R × Y → X has the property that G(λ, ·) = ∇X g(λ, ·)

where g : R × Y → R is C 2 , then the equation G(λ, y) = 0 is said to be the Euler-Lagrange equation of the functional g. Equivalently we say that the equation has gradient structure. In the theory of nonlinear equations, those with gradient structure have important special properties. We return to this in §11.3.

3.5 INVERSE AND IMPLICIT FUNCTION THEOREMS The inverse function theorem says that if the Fr´echet derivative of a nonlinear operator F at x0 is invertible, then the nonlinear operator itself is a bijection from an open neighbourhood of x0 onto an open neighbourhood of F (x0 ). Before giving its precise statement we prove a technical result. L EMMA 3.5.1 Let X and Y be Banach spaces, U ⊂ X a convex open set, and let F : U → Y be Fr´echet differentiable at each point of U . Suppose that there exists A ∈ L(X, Y ) such that dF [x] − A ≤ M for all x ∈ U . Then for all x1 , x2 ∈ U ,

F (x1 ) − F (x2 ) − dF [x2 ](x1 − x2 ) ≤ 2M x1 − x2 . Proof. Let x2 ∈ U be arbitrary, but fixed. We apply the Lemma 3.1.4 to G defined on U by G(x) = F (x) − dF [x2 ]x, x ∈ U . Then

dG[x] = dF [x] − dF [x2 ] ≤ dF [x] − A + A − dF [x2 ] ≤ 2M. Therefore, by Lemma 3.1.4,

F (x1 ) − F (x2 ) − dF [x2 ](x1 − x2 )

= G(x1 ) − G(x2 ) ≤ 2M x1 − x2 . This proves the lemma. T HEOREM 3.5.2 (Inverse Function Theorem) Let x0 ∈ U, an open subset of a Banach space X, and let F ∈ C 1 (U, Z) where Z is also a Banach space. Suppose that dF [x0 ] ∈ L(X, Z) is a homeomorphism. Then there exists a connected open set U0 ⊂ U with x0 ∈ U0 and an open ball W ⊂ Z with F (x0 ) ∈ W , such that F : U0 → W is a bijection and F −1 ∈ C 1 (W, X). If, in addition, F ∈ C k (U, Z), k ∈ N, then F −1 ∈ C k (W, X).  R EMARK 3.5.3 We say that F U is a diffeomorphism onto W . 0

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CHAPTER 3

  Proof. By considering the mapping x → dF [x0 ]−1 F (x0 + x) − F (x0 ) in place of F , there is no loss of generality in supposing that X = Z, that x0 = F (x0 ) = 0 ∈ X, and that dF [0] = I ∈ L(X, X), the identity operator. From this special case the general result follows. In the new setting, let r ∈ (0, 1) be chosen so that if x ≤ r then x ∈ U and dF (x) − I < 1/4. There is no further loss of generality in supposing that U = {x : x < r}. We will now show that, if y ∈ X is sufficiently close to zero, a sequence {xn } ⊂ U can be defined inductively by the recipe x0 = 0, xn+1 = y + xn − F (xn ), and the sequence so defined converges to a solution x of the equation F (x) = y. Now, provided that xn ∈ U for all n ∈ N with n ≤ k, the definition of xn and Lemma 3.1.4 give that x1 = y and,

xn+1 − xn = (F (xn−1 ) − xn−1 ) − (F (xn ) − xn )

≤ xn − xn−1 sup dF [xn−1 + t(xn − xn−1 )] − I

0≤t≤1

1 ≤ xn − xn−1 , 4 for all 1 ≤ n ≤ k. Now choose y with y < 3r/4. It can be seen, by induction, that for all n ≥ 0,

xn+1 − xn ≤ 4−n y and xn ≤ 4 y /3 < r. Moreover, for all m > n

xn − xm ≤

m−1 

xk+1 − xk ≤ 4−(n−1) y

k=n

which converges to 0 as n → ∞ . Therefore {xn } is a Cauchy sequence which converges in the Banach space X, to x, say, and x ≤ 4 y /3 < r. By continuity, x = y + x − F (x), whence F (x) = y. Let W = {y ∈ X : y < 3r/4}, U0 = {x : x < r and F (x) ∈ W }. Then U0 is open, W is an open ball, and F : U0 → W is a bijection. Suppose that y1 , y2 ∈ W , x1 , x2 ∈ U0 , F (x1 ) = y1 and F (x2 ) = y2 . Then

y2 − y1 = F (x2 ) − F (x1 )

= (x2 − x1 ) + (dF [x2 ] − I)(x2 − x1 )   + F (x2 ) − F (x1 ) − dF [x2 ](x2 − x1 )

≥ (x2 − x1 ) − (dF [x2 ] − I)(x2 − x1 )

− (F (x2 ) − F (x1 ) − dF [x2 ](x2 − x1 ) ≥

(3.3)

1

x2 − x1 , 4

(3.4)

CALCULUS IN BANACH SPACES

37

because of the choice of r and Lemma 3.5.1. This shows that F −1 is Lipschitz continuous on W . Hence U0 = F −1 (W ) ⊂ U is connected. Now we show that F −1 is differentiable at each point y ∈ W and that the derivative d(F −1 )[y] depends continuously on y ∈ W . Let y ∈ W and let k ∈ Y be such that y + k ∈ W . Suppose that F (x) = y and F (x + h) = y + k, where x, x + h ∈ U0 . Since x < r it follows that dF [x] = dF [F −1 (y)] is a homeomorphism (Lemma 2.5.1) . Therefore  −1 k

F −1 (y + k) − F −1 (y) − dF [F −1 (y)]

k

 −1 (F (x + h) − F (x))

(x + h) − x − dF [x] =

k

 −1 {F (x + h) − F (x) − dF [x]h} h

dF [x] · =

h

k

 −1 F (x + h) − F (x) − dF [x]h h

≤ dF [x] · →0

h

k

as k → 0 by the definition of dF [x] and the fact that 4 k ≥ h , which follows from (3.4). This shows that, for all y ∈ W , F −1 is differentiable at y with  −1 . d(F −1 )[y] = dF [F −1 (y)] The continuity of d(F −1 ) is immediate from Lemma 2.5.1. Finally, from the formula for d(F −1 )[y] and the chain rule it follows that if F ∈ C k (U, X) then F −1 ∈ C k (W, X). The next result is a corollary of the inverse function theorem. T HEOREM 3.5.4 (Implicit Function Theorem) Let X, Y and Z be Banach spaces and suppose that U ⊂ X × Y is open. Let (x0 , y0 ) ∈ U. Suppose also that, for some k ∈ N, F ∈ C k (U, Z), that F (x0 , y0 ) = z0 , and that the partial derivative ∂x F [(x0 , y0 )] ∈ L(X, Z) is a homeomorphism. Then there exists an open ball V ⊂ Y with centre y0 ∈ Y , a connected open set W ⊂ U and a mapping φ ∈ C k (V, X) such that (x0 , y0 ) ∈ W and F −1 (z0 ) ∩ W = {(φ(y), y) : y ∈ V }. Proof. Define a new function G ∈ C k (U, Z × Y ) by   G(x, y) = F (x, y), y . Clearly G(x0 , y0 ) = (z0 , y0 ) and   dG[(x0 , y0 )](x, y) = ∂x F [(x0 , y0 )]x + ∂y F [(x0 , y0 )]y, y for (x, y) ∈ X × Y. Therefore dG[(x0 , y0 )] has a bounded inverse dG[(x0 , y0 )]−1 : Z × Y → X × Y given by     (dG[(x0 , y0 )])−1 (z, y) = (∂x F [(x0 , y0 )])−1 z − ∂y F [(x0 , y0 )]y , y

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CHAPTER 3

for (x, y) ∈ X × Y, and one may apply the inverse function theorem 3.5.2 to G to obtain an open connected set W ⊂ U and an open ball R ⊂ Z × Y such that (x0 , y0 ) ∈ W , (z0 , y0 ) is the centre of R, and G : W → R is a diffeomorphism of class C k (W, R) for the same k as in the statement of the theorem. It suffices now to put V = {y : (z0 , y) ∈ R} and to say that φ(y) = x for y ∈ V if and only if G−1 (z0 , y) = (x, y) ∈ W . Then (x, y) ∈ W and F (x, y) = z0 if and only if (z0 , y) ∈ R and G−1 (z0 , y) = (x, y) ∈ W . Now G−1 is of class C k on R implies that y → G−1 (z0 , y) is of class C k on V . Let P denote the projection of X × Y onto X × {0}. Then, since (φ(y), 0) = P (G−1 (z0 , y)), it follows that φ is of class C k on V . This completes the proof. D EFINITION 3.5.5 A set M ⊂ Fn is called an m-dimensional manifold of class C k , or a C k -manifold, k ≥ 1, if, for all points x ∈ M , there is an open neighbourhood Ux of 0 ∈ Fm and a function f : Ux → M such that f (0) = x, df [0] is a finite-dimensional linear transformation of rank m and f maps open sets in U onto relatively open sets in M . R EMARK 3.5.6 Suppose that a mapping F : Fn × Fm → Fn is of class C k with F (x0 , y0 ) = z0 and that ∂x F [(x0 , y0 )] is a bijection on Fn . Then the implicit function theorem 3.5.4 defines a C k -manifold of dimension m by the equation F (x, y) = z0 for (x, y) in a neighbourhood of (x0 , y0 ).

3.6 PERTURBATION OF A SIMPLE EIGENVALUE The following corollary of the implicit function theorem is often useful even in finite-dimensional linear algebra. The notation is that of Definition 2.7.8 and Lemma 2.7.9. P ROPOSITION 3.6.1 Let X ⊂ Y be Banach spaces and let the embedding operator ι ∈ L(X, Y ). Let s → L(s) be a mapping of class C k , k ≥ 1, from (−1, 1) into L(X, Y ). Suppose that µ0 is a simple eigenvalue of L(0) with eigenvector ξ0 ∈ X where

ιξ0 Y = 1. Then there exists > 0 and a C k -curve {(µ(s), ξ(s)) : s ∈ (− , )} ⊂ R × X such that (µ(0), ξ(0)) = (µ0 , ξ0 ), L(s)ξ(s) = µ(s)ι ξ(s) and ξ(s) = ξ0 + η(s), where ιη(s) ∈ range (L(0) − µ0 ι). Moreover, µ(s) is a simple eigenvalue of L(s) and if |s| < and µ is an eigenvalue of L(s) with |µ0 − µ| < then µ = µ(s). Proof. Define a mapping G : R × X × (−1, 1) → Y × R by   G(µ, x, s) = µι x − L(s)x, y ∗ (ιx) − 1 ,

CALCULUS IN BANACH SPACES

39

∗ ∗ where using Corollary 2.3.1 (b) so that y ∗ (ιξ0 ) = 1 and  y ∈ Y is chosen  ∗ y range (µ0 ι − L(0)) = 0. Then G(µ0 , ξ0 , 0) = (0, 0) and

  ∂(µ,x) G[(µ0 , ξ0 , 0)](µ, x) = µι ξ0 + µ0 ιx − L(0)x, y ∗ (ιx) for (µ, x) ∈ R × X. Suppose that ∂(µ,x) G[(µ0 , ξ0 , 0)](µ, x) = (0, 0). Then µι ξ0 ∈ range (µ0 ι − L(0)) and, since µ0 is a simple eigenvalue of L(0), µ = 0. Therefore (L(0) − µ0 ι)x = 0. Since µ0 is simple, x ∈ span {ξ0 } and, since y ∗ (ιx) = 0 it follows that x = 0. Thus ∂(µ,x) G[(µ0 , ξ0 , 0)] is injective. The surjectivity of ∂(µ,x) G[(µ0 , ξ0 , 0)] follows immediately from the fact that, since µ0 is a simple eigenvalue of L(0), Y = range (µ0 ι − L(0)) ⊕ span {ιξ0 }. Since ∂(µ,x) G[(µ0 , ξ0 , 0)] is therefore a bijection, it is a homeomorphism by Corollary 2.4.3 and the implicit function theorem 3.5.4 gives the existence of a C k curve {(µ(s), ξ(s)) : s ∈ (− , )} ⊂ R × X of solutions of G(µ, x, s) = (0, 0) with (µ(0), y(0))) = (µ0 , ξ0 ). Since y ∗ (ιξ(s)) = 1, the choice of y ∗ gives that ξ(s) = ξ0 + η(s) where ιη(s) ∈ range (µ0 ι − L(0)). The result of the proposition will have been proven once it is shown that each µ(s) is a simple eigenvalue of L(s). It follows from Proposition 2.7.7 that, for s sufficiently small, µ(s)ι − L(s) is a Fredholm operator of index zero. Suppose that for s ∈ (− , ) there exists x(s) such that (µ(s)ι − L(s))x(s) = 0, x(s) X = 1.  −1 Let x(s) range (µ0 ι−  = α(s)ξ0 +z(s) where, as in Lemma 2.7.9, z(s) ∈ X1 = ι L(0)) and without loss of generality suppose α(s) ≥ 0. Then (µ0 ι − L(0))z(s) = (µ0 ι − L(0))x(s)

 = (µ0 ι − µ(s))ι − (L(0) − L(s)) x(s) → 0 in Y as s → 0. ˆ(s) = Therefore z(s) → 0 in X, and so α(s) → ξ0 −1 X , as s → 0. Now let x x(s)/α(s). Then (µ(s), x ˆ(s)) → (µ0 , ξ0 ) in R × X and G(µ(s), x ˆ(s), s) = 0. The implicit function theorem 3.5.4 implies that, for s sufficiently small, x(s) is a scalar multiple of ξ(s). This shows that ker(µ(s)ι − L(s)) = span {ξ(s)}. Since ξ(s) → ξ0 in X and ξ0 ∈ / X1 = ι−1 (Y1 ), which is closed in X, it follows that ξ(s) ∈ / X1 for all s sufficiently small. Therefore X = X1 ⊕ span {ξ(s)} for s sufficiently small. Finally suppose that for s ∈ (− , ) there exists p(s) ∈ X such that (µ(s)ι − L(s))p(s) = ιξ(s).

40

CHAPTER 3

By the preceding paragraph there is no loss in assuming that p(s) ∈ X1 . If

p(sn ) X → ∞ for a sequence sn → 0, it is immediate that   µ0 ι − L(0)

p(sn )  → 0 in Y

p(sn ) X

as n → ∞. Therefore p(sn )/ p(sn ) X → 0 in X, by Lemma 2.7.9. But this is false. Hence p(s) X is bounded for s sufficiently small. Therefore

   µ0 ι − L(0) p(s) = (µ0 − µ(s))ι − (L(0) − L(s)) p(s) + ιξ(s) → ιξ0 in Y . Since Y1 = range (µ0 ι − L(0)) is closed in Y , ιξ0 ∈ Y1 . This contradiction proves that µ(s) is a simple eigenvalue of L(s).

3.7 NOTES ON SOURCES Calculus in Banach spaces is covered in many textbooks, for example Cartan [16], Dieudonn´e [28] and Schwartz [52].

Chapter Four Multilinear and Analytic Operators The theory of higher order Fr´echet derivatives leads to the notion of multilinear operators. The question of whether the Taylor polynomials of F (see Definition 3.3.2) converge to F (x) for some x ∈ X as n → ∞ leads to the theory of analytic functions.

4.1 BOUNDED MULTILINEAR OPERATORS Suppose that Y and X1 , · · · , Xp , p ∈ N, are Banach spaces over F. A mapping m : X1 × · · · × Xp → Y is said to be a multilinear operator, in this case plinear, if it is linear in each variable separately, that is, for all k ∈ {1, · · · , p} and xj ∈ Xj , j = k, x → m(x1 , · · · , xk−1 , x, xk+1 , · · · , xp ) is linear in x over F. It is said to be a bounded multilinear operator if, in addition, sup{ m(x1 , x2 , · · · , xp ) : x1 , · · · , xp ≤ 1} = M < ∞.

(4.1)

If m is multilinear and xj = 0 for some j, then m(x1 , · · · , xp ) = 0. Otherwise, if xj = 0 for all j and m is bounded, we find that    x x2 xp   1 , ,··· ,  ≤ M, m

x1 x2

xp

whence

m(x1 , · · · , xp ) ≤ M x1

x2 · · · xp . The proofs of the next few propositions are so similar to those for L(X, Y ) that we omit them. P ROPOSITION 4.1.1 Suppose that m : X1 × · · · × Xp → Y is a multilinear operator. Then the following are equivalent statements. m : X1 × · · · × Xp → Y is continuous; m is continuous at (0, · · · , 0) ∈ X1 × · · · × Xp ; m is bounded.

42

CHAPTER 4

A bounded multilinear operator is called symmetric if m(x1 , · · · , xp ) = m(xπ(1) , · · · , xπ(p) ) for all π ∈ Sp , the symmetric group. P ROPOSITION 4.1.2 The set M(X1 , · · · , Xp ; Y ) of bounded multilinear operators endowed with the norm

m = sup{ m(x1 , x2 , · · · , xp ) : x1 , · · · , xp ≤ 1} is a Banach space in which the symmetric operators form a closed subspace. When Xk = X, 1 ≤ k ≤ p, we abbreviate M(X1 , · · · , Xp ; Y ) as Mp (X, Y ). E XAMPLE 4.1.3 Fr´echet derivatives give an important class of multilinear operators. If F : U ⊂ X → Y has a k th Fr´echet derivative at x0 ∈ U , it is clear (Proposition 3.2.3) that (x1 , · · · , xk ) → dk F [x0 ](x1 , · · · , xk ) is a bounded, symmetric k-linear operator. That dk F : U → Mk (X, Y ) is continuous is equivalent to saying that F ∈ C k (U, Y ). E XAMPLE 4.1.4 When X = Y = F, it follows from the Riesz representation theorem 2.3.3 that every element mp of Mp (X, Y ) is given by mp (x1 , · · · , xp ) = Ap x1 · · · xp  product

for some Ap ∈ F, and therefore all mp are symmetric in this case. E XAMPLE 4.1.5 (Determinants) An n × n matrix A has rows (ai1 , · · · , ain ) ∈ Fn , 1 ≤ i ≤ n. Its determinant det A, defined by det A =



σ(π)

π∈Sn

n 

(4.2)

aiπ(i) ,

i=1

where σ(π) denotes the of π ∈ Sn (the symmetric group), is therefore an n  signature n-linear function on Fn . Since interchanging two rows of A changes the sign of det A, the determinant is not a symmetric operator on the rows of A. Instead we say that it is skew-symmetric. As a consequence, the determinant of a matrix with two equal rows is zero. A function h on a Banach space X is called p-homogeneous if f (αx) = αp f (x) for all α ∈ F and x ∈ X. Thus A → det A is n-homogeneous on the space of n × n matrices. Let X j denote the product of j copies of the same space X. When m ∈ Mp (X, Y ) is symmetric, define a mapping on X j by (x1 · · · , xj ) → m(x1 , · · · , x1 , x2 , · · · , x2 , · · · , xj , · · · , xj )    k1 times

=

m xk11

k · · · xj j ,

k2 times

kj times

(4.3)

where j ∈ N and k1 + · · · + kj = p. (This defines the notation on the second line.)

43

MULTILINEAR AND ANALYTIC OPERATORS

P ROPOSITION 4.1.6 (a) Suppose m ∈ M(X1 , · · · , Xp ; Y ). Then m is infinitely differentiable on the product space X1 × · · · × Xp and its first derivative at a point (x1 , · · · , xp ) ∈ X1 × · · · × Xp is given by dm[(x1 , · · · , xp )](z1 , · · · , zp ) =

p 

m(x1 , · · · , xk−1 , zk , xk+1 , · · · , xp ),

k=1

for all (z1 , · · · , zp ) ∈ X1 × · · · × Xp . (b) If m ∈ Mp (X, Y ) is symmetric then the mapping, h say, from X j to Y defined by (4.3) is infinitely differentiable and for (y1 , · · · , yj ) ∈ X j , dh[(x1 , · · · , xj )](y1 , · · · , yj ) =

j 

(l−1) kl −1 (l+1) kl mxk11 · · · xl−1 xl xl+1 · · · xj j yl .

k

k

k

l=1

Proof. The proof, which is almost obvious and complicated only by notation, is left as an exercise. R EMARKS 4.1.7

The mapping (l−1) kl −1 (l+1) ul → mxk11 · · · xl−1 xl xl+1 · · · xj j ul

k

k

k

(l−1) kl −1 (l+1) belongs to L(X, Y ) and so mxk11 · · · xl−1 xl xl+1 · · · xj j may be identified with an element of L(X, Y ), which yields a shorthand notation for the partial derivative of h in part (b),

k

k

k

(l−1) kl −1 (l+1) ∂xl h[(x1 , · · · , xj )] = kl mxk11 · · · xl−1 xl xl+1 · · · xj j .

k

P ROPOSITION 4.1.8

k

k

Suppose that m ∈ Mp (X, Y ) and define F : X → Y by F (x) = m(x, · · · , x), x ∈ X.

Then dk F [0] = 0 except when k = p and dp F [0](x1 , x2 , · · · , xp ) =



m(xπ(1) , xπ(2) , · · · , xπ(p) ),

π∈Sp

where Sp is the symmetric group. Proof. This is an elementary exercise using induction on the order of differentiation.

44

CHAPTER 4

` DE BRUNO FORMULA 4.2 FAA Although the formula for the k th derivative of the composition of two C k -functions looks intimidating, its derivation is straightforward except that the notation is elaborate. Let X, Y and Z be Banach spaces with U ⊂ X and V ⊂ Y open sets. Suppose that F ∈ C k (U, Y ) maps U into V , G ∈ C k (V, Z) and H ∈ C k (U, Z) is defined by H(x) = G(F (x)), x ∈ U. The formula for dk H is most easily expressed in terms of some auxiliary notation. Let (n+) denote the set of n-tuples of positive integers. For β = (β1 , · · · , βn ) ∈ (n+), let β! = β1 !β2 ! · · · βn ! and |β| = β1 + · · · + βn . For x0 ∈ U , β ∈ (n+), (x1 , · · · , xk ) ∈ X k , let Dnk (F, G)[x0 ] : X k → Y be defined by Dnk (F, G)[x0 ](x1 , · · · , xk ) =

 1  dn G[F (x0 )] dβ1 F [x0 ](xσ(1) , · · · , xσ(β1 ) , β! σ∈Sk β∈(n+) |β|=k

  dβ2 F [x0 ] xσ(β1 +1) , · · · , xσ(β1 +β2 ) , · · ·   · · · , dβn F [x0 ] xσ(β1 +β2 +···+βn−1 +1) , · · · , xσ(k) ,

where Sk is the symmetric group. T HEOREM 4.2.1

(Fa`a de Bruno Formula) For x0 ∈ U and k ∈ N,

dk H[x0 ](x1 , · · · , xk ) =

k  1 k Dn (F, G)[x0 ](x1 , · · · , xk ). n! n=1

Proof. Since both F and G have k th Fr´echet derivatives, so has H. From the Chain rule 3.1.3, it is obvious that the k th derivative of H at x0 depends only on the first k derivatives of F at x0 and of G at F (x0 ). Therefore it suffices to replace F by its k th order Taylor polynomial at x0 and G by its k th order Taylor polynomial at F (x0 ), and without loss of generality we may assume that x0 = 0 ∈ X, F (x0 ) = F (0) = 0 ∈ Y and G(0) = 0 ∈ Z. Thus henceforth F (x) =

k k   1 1 fm xm and G(y) = gn y n m! n! m=1 n=1

where fm = dm F [0] and gn = dn G[0] are bounded symmetric m-linear operators.

45

MULTILINEAR AND ANALYTIC OPERATORS

Therefore k k n  1  1 H(x) = gn fm xm n! m! n=1 m=1

=

k k k k     1  1 1 1 gn fβ1 xβ1 , fβ2 xβ2 , · · · , fβn xβn . n! β1 ! β2 ! βn ! n=1 β1 =1

β2 =1

βn =1

By Proposition 4.1.8, in this expression for H(x) the k-linear term k   1  1  1 1 gn fβ1 xβ1 , fβ2 xβ2 , · · · , fβn xβn n! β1 ! β2 ! βn ! n=1 β∈(n+) |β|=k

=

k    1  1 n d G[F (x0 )] dβ1 F [x0 ]xβ1 , · · · , dβn F [x0 ]xβn , n! β! n=1 β∈(n+) |β|=k

is a function of x ∈ X whose k th derivative everywhere coincides with the k th derivative of H at 0. A further application of Proposition 4.1.8 now leads to the Fa`a de Bruno formula. This completes the proof.

4.3 ANALYTIC OPERATORS Now we embark on a study of power series and F- analytic functions in Banach spaces. Let X and Y be Banach spaces over F. Let U be an open subset of X. D EFINITION 4.3.1 A mapping F : U → Y is F- analytic at x0 ∈ U if, for all x ∈ U with x − x0 sufficiently small, F (x) =

∞ 

mk (x − x0 )k

(4.4)

k=0

where F (x0 ) = m0 (x − x0 )0 = m0 ∈ Y , mk ∈ Mk (X, Y ) is symmetric and there exists r > 0 such that sup rk mk = M < ∞.

(4.5)

k≥0

The series on the right in (4.4) is a power series in x − x0 . The function F is said to be F-analytic on U if it is F-analytic at every point of U . (When it does not matter we will omit F and speak of analytic functions.) The expressions analytic mapping, analytic operator and analytic function will be used interchangeably in what follows.

46

CHAPTER 4

Because of (4.5), for all x ∈ X with x − x0 < r, ∞ 

mk (x − x0 )k ≤ M

k=0

∞ 

x − x0 k k=0

rk

=

Mr 0, all depending on x0 , such that  k  d F [x] ≤ C k! for all x ∈ U with x − x0 < r. Rk

(4.7)

Then F is analytic on U . R EMARK 4.3.3

Since Stirling’s formula says that √ 2πn nn lim = 1, n→∞ en n!

(4.8)

(4.7) is equivalent to k  k  d F [x] ≤ C k for all x ∈ U with x − x0 < r, Rk

for different constants C and R. (The inequality en ≥ nn /n! is sufficient for this observation.)

47

MULTILINEAR AND ANALYTIC OPERATORS

Proof. Let x0 , x ∈ U . Then in (4.6),

x − x0 n+1 sup dn+1 F [(1 − t)x0 + tx]

(n + 1)! 0≤t≤1 C ≤ n+1 x − x0 n+1 → 0 as n → ∞ R

Rn (x, x0 ) ≤

if x − x0 < min{r, R}. This proves the result. We aim to prove a converse of Theorem 4.3.2, but first some observations. Differentiating the identity ∞

 1 xk , |x| < 1, = 1−x k=0

p times yields the new identity

1 p+1  = 1−x ∞



k=0

k+p p

 xk , |x| < 1, p ∈ N.

(4.9)

Suppose that r > 0 and {mk } is a sequence of symmetric k-linear operators on X with rk mk ≤ M for all k ∈ N. Let x, z1 , · · · , zp ∈ X with x < (1 − )r,

∈ (0, 1), and note that, by (4.9),  ∞   k+p

mk+p xk z1 · · · zp

p k=0  ∞   k+p ≤

mk+p x k z1 · · · zp

p k=0  ∞ 

z1 · · · zp  k + p x k ≤M p rp r k=0 

p+1 r

z1 · · · zp

=M rp r − x

z · · · z  1 p . ≤ −1 M

p rp Hence, when x < (1 − )r, a symmetric operator Mpx ∈ Mp (X, Y ) may be defined at (z1 , · · · , zp ) ∈ X p by Mpx (z1 , · · ·

 ∞   k+p mk+p xk z1 · · · zp ∈ Y , zp ) = p k=0

and

Mpx ≤

M rp p+1

.

(4.10)

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CHAPTER 4

Also, if x < (1 − )r and z < r, then  ∞  ∞  M  z p k+p k p

mk+p x z ≤ < ∞, p

p=0 r

(4.11)

k,p=0

and so the series

 ∞   k+p mk+p xk z p p

k,p=0

is summable in norm, and therefore convergent to a sum which is independent of the order in which summation is done. This leads to the following result in which (4.13) is a converse of Theorem 4.3.2. P ROPOSITION 4.3.4 Let (4.5) hold and let F be defined by (4.4). Then F is analytic at every point x of the set U0 = {x ∈ U : x − x0 < r}, F ∈ C ∞ (U0 , Y ) and mk =

dk F [x0 ] , for all k ≥ 0. k!

Also the pth derivative, dp F , is analytic at x for all p ∈ N and x − x0 < r, dp F [x](x1 , · · · , xp ) =

∞  (k + p)! k=0

k!

mk+p (x − x0 )k x1 x2 · · · xp ,

(4.12)

and there exists C > 1, R ∈ (0, 1) such that if x − x0 ≤ 12 r, p! for all p ∈ N. (4.13) Rp If K ⊂ U is a compact set, then there exists C and R such that (4.13) holds for all x ∈ K.

dp F [x] ≤ C

Proof. We will show first that if ˆ x − x0 < r then F is analytic at x ˆ. This follows from (4.11) since F (x) =

∞ 

ml (x − x0 )l =

l=0

∞ 

 l ˆ) + (ˆ x − x0 ) ml (x − x

l=0

 l  ∞   l ml (x − x = ˆ)l−k (ˆ x − x0 )k k l=0 k=0  ∞  ∞   l ml (x − x = ˆ)l−k (ˆ x − x0 )k k k=0 l=k  ∞  ∞   k+p mp+k (x − x = ˆ)p (ˆ x − x0 )k k =

k=0 p=0 ∞  Mpxˆ−x0 (x p=0

−x ˆ)p ∈ Y.

(4.14)

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MULTILINEAR AND ANALYTIC OPERATORS

Since (4.10) holds, this shows that F is analytic at x ˆ provided ˆ x − x0 < r. Next, observe that when x − x0 < r, ∞  

F (x) − F (x0 ) − m1 (x − x0 ) =  ml (x − x0 )l  = o( x − x0 ) l=2

as x − x0 → 0, by (4.5). Therefore F is Fr´echet differentiable at x0 and dF [x0 ]x = m1 x for all x ∈ X. By the same token it follows from (4.14) that when ˆ x−x0 < r, F is differentiable at x ˆ and (ˆ x−x0 )

dF [ˆ x]x = M1

x=

∞ 

(k + 1)mk+1 (ˆ x − x0 )k x.

k=0

Thus dF is analytic at x0 since, for x with x − x0 sufficiently small, dF [x] =

∞ 

(k + 1)mk+1 (x − x0 )k

k=0

where the right-hand side is regarded as an element of L(X, Y ) expressed as a power series. (To confirm that (4.5) holds here note that (k + 1) mk+1 ≤ M (k + 1)rk+1 ≤ {M/r}(2/r)k . Now replace r with r/2 and M with M/r.) It now follows by induction that all the derivatives of F exist at every point of x with x − x0 < r and are given by a power series in x − x0 . The formula (4.12) also follows by induction. Therefore for p ∈ N and x ∈ X with x − x0 ≤ 12 r,

dp F [x] ≤ ≤

∞  (k + p)! k=0 ∞  k=0

k!

mk+p x − x0 k

∞ (k + p)! M r k p!  (k + p)!  1 k = M k! rk+p 2 rp k! p! 2 k=0

2p p! = 2M p by (4.9). r Finally, if K ⊂ U is compact then (4.13) holds uniformly for x ∈ K, since every open cover of K has a finite sub-cover. This completes the proof. Here are important, non-trivial examples of analytic operators. E XAMPLE 4.3.5 (Operator Inverses) Suppose T ∈ L(X, Y ) is a bijection. Then, by Lemma 2.5.1, there exists > 0 such that if L ∈ L(X, Y ) and L−T
1 and R ∈ (0, 1), given by the last part of Proposition 4.3.4, be such that

f k [ξ] ≤

Ck! for ξ ∈ FN with |ξ| ≤ M + 1. Rk

(4.15)

Now for u ∈ X with u ≤ M + 1, t ∈ [0, 1] and n ∈ N let fn (u(t)) =

n  f (k) (u0 (t)) k=0

k!

(u(t) − u0 (t))k .

By Taylor’s theorem, f (u(t)) − fn (u(t)) = Rn (u(t), u0 (t)) where

Rn (u(t), u0 (t))

≤

u(t) − u0 (t) k+1 sup f (k+1) ((1 − s)u0 (t) + su(t)) . (k + 1)! 0≤s≤1

It follows from (4.15) that fn (u) → f (u) in the Banach space X as n → ∞ provided that u − u0 < R. It remains to define symmetric mk ∈ Mk (X, X) by mk (v1 , · · · , vk )(t) =

 dk f [u0 (t)]  v1 (t), v2 (t), · · · , vk (t) k!

so that F (u) =

∞  k=0

mk (u − u0 )k ,

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MULTILINEAR AND ANALYTIC OPERATORS

  where the series on the right converges in C [0, 1], FM . Therefore the Nemytskii operator is F-analytic on X. A similar argument leads to the same conclusion for  the same Nemytskii operator regarded as a mapping on C n [0, 1], FN , and on other Banach spaces of functions as well. R EMARK 4.3.7 (Analytic Operators on Lebesgue Spaces) Suppose P is a polynomial of degree p and P ◦ u ∈ Lq [0, 1] for all u ∈ Lr [0, 1]. It is easy to see that pq ≤ r. Consider the mapping f (u) = up where p ∈ N and pq ≤ r. It follows that the th k -derivative, k ≤ p, of f at u0 ∈ Lr [0, 1] is given by dk f [u0 ](u1 , · · · , uk ) =

p! up−k u1 · · · uk , (p − k)! 0

which, by H¨older’s inequality, is a bounded k-linear operator on Lr [0, 1]. All higher derivatives of f are zero. The Nemytskii operator from Lr [0, 1] to Lq [0, 1] so defined is then F-analytic. For example u → u2 is analytic from L3 [0, 1] to L1 [0, 1]. (However there are no non-trivial F-analytic operators from Lp [0, 1] to itself, 1 ≤ p < ∞, as Example 3.1.10 shows.) It is well known that if U ⊂ C is open and connected, and f : U → C is a nonconstant C-analytic function, then the zero set of f has no limit points in U . This is clearly false for C-analytic functions from C2 into C2 , as the following example shows. E XAMPLE 4.3.8

Let F : C2 → C2 be defined by F (x, y) = (xy, xy)

for all (x, y) ∈ C2 . Then F is C-analytic because its Taylor series has one term: F (x, y) = m2 (x, y)2 where, for (x1 , y1 ), (x2 , y2 ) ∈ C2 ,   m2 (x1 , y1 ), (x2 , y2 ) = 12 (x1 y2 + x2 y1 , x1 y2 + x2 y1 ). However F (x, y) = (0, 0) when xy = 0. Thus every point of the zero set of F is a limit point of the zero set. However we have the following result. T HEOREM 4.3.9 Suppose that X and Y are Banach spaces, that U ⊂ X is an open connected set and that F : U → Y is F-analytic. Suppose also that there is a non-empty open set W ⊂ U on which F is identically 0. Then F is identically zero on U . Proof. Let V ⊂ U be the set of points x ∈ U at which all the derivatives of F are zero. The set V is non-empty since, by hypothesis, W ⊂ V . Since F ∈ C ∞ (U, Y ), V is an intersection of sets which are closed in U and so V is closed in U . Since F is an F-analytic function, Proposition 4.3.4 implies that V is open, and therefore open in U . But U is connected, so U = V . This completes the proof.

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CHAPTER 4

4.4 ANALYTIC FUNCTIONS OF TWO VARIABLES Let X, Y and Z be Banach spaces and U an open subset of X × Y . Let (x0 , y0 ) ∈ U and let F : U → Z be analytic at (x0 , y0 ). In other words, for (x, y) sufficiently close to (x0 , y0 ), F (x, y) =

∞ 

mk (x − x0 , y − y0 )k

(4.16)

k=0

where mk ∈ Mk (X × Y, Z) is symmetric with supk≥0 rk mk < ∞ for some r > 0. Now let Mp, q (X, Y ; Z) = M(X, · · · , X , Y, · · · , Y ; Z),   p times

and define mp,q ∈ M

p,q

q times

(X, Y ; Z) by

mp,q (x1 , · · · , xp , y1 , · · · , yq )

  = mp+q (x1 , 0), · · · , (xp , 0), (0, y1 ), · · · , (0, yq ) .

It follows that mp,q ≤ mp+q and, since (x, y) = (x, 0) + (0, y), an argument similar to that for (4.14) now yields that  (p + q)! mp,q ((x − x0 )p , (y − y0 )q ), p! q! p, q≥0   sup mp, q rp+q < ∞,

F (x, y) =

(4.17) (4.18)

p, q≥0

where r > 0. Note that mp,q is symmetric separately in the first p, and in the last q, variables. The series  (p + q)! mp,q ((x − x0 )p , (y − y0 )q ) p! q! p, q≥0

is summable in norm for x − x0 + y − y0 < r. It follows that mp,q =

∂xp ∂yq F [(x0 , y0 )] (p + q)!

where ∂xp ∂yq F [(x0 , y0 )] ∈ Mp,q (X, Y ; Z), and hence F (x, y) =

 ∂xp ∂yq F [(x0 , y0 )] ((x − x0 )p , (y − y0 )q ), p! q!

(4.19)

p, q≥0

 ∂xp ∂yq F [(x0 , y0 )] rp+q  < ∞, (p + q)! p, q≥0 sup

(4.20)

for some r > 0. Conversely, (4.17) and (4.18) defines an F-analytic mapping, in the sense of Definition 4.3.1, which satisfies (4.19) and (4.20).

53

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4.5 ANALYTIC INVERSE AND IMPLICIT FUNCTION THEOREMS We begin by proving the analytic version of the inverse function theorem. Note that the proof here does not assume that the composition of two analytic functions is analytic; instead this emerges as a corollary of the theorem. Let X be a Banach space and let Br (X), r ∈ (0, 1), denote the ball of radius r centred at 0 ∈ X. For r ∈ (0, 1), let Br = Br2 (X) × Br (X) ⊂ X × X and let (Er , · r ) denote the Banach space of functions u which are analytic from Br into X with u(x, y) =



um,n xm y n , (x, y) ∈ X × X,

m, n≥0

where 

u r =

um,n r2m+n < ∞.

m, n≥0

Note that the norm · r uses different weights for the x and y dependence of functions in Er . However, since (4.16) implies (4.17) and (4.18), any function F which maps a neighbourhood of the origin in X × X analytically into X belongs to Er for all r > 0 sufficiently small. That Er is a Banach space follows by a proof, almost identical to that of Proposition 5.1.3 (below), in which the completeness of F is replaced with the completeness (Proposition 4.1.2) of the space of k-linear operators on X for all k ∈ N0 . Let Fr denote the closed subspace of Er of functions u ∈ Er with u(x, y) =



um,n xm y n .

m≥0, n≥1

Define L ∈ L(Fr , Fr ) by Lu (x, y) =

 m≥0, n≥1

um,n m n x y , (x, y) ∈ Br , u ∈ Fr . n

Clearly L = 1. Now for w ∈ Er arbitrary but fixed define Lw u for u ∈ Fr by Lw u (x, y) = ∂y u[(x, y)]w(x, y) − ∂y u[(x, 0)]w(x, 0), (x, y) ∈ Br . L EMMA 4.5.1 (a) The operator Lw ◦ L ∈ L(Fr , Fr ) and Lw ◦ L ≤ w r /r. (b) Let w0 ∈ Er be defined by w0 (x, y) = y, (x, y) ∈ Br . Then Lw0 ◦ L is the identity on Fr .

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Proof. Let w(x, y) =

 p, q≥0

wp,q xp y q . Then for u ∈ Fr ,

Lw ◦ Lu (x, y)

  

   = um,n+1 xm y n wp,q xp y q − um,1 xm wp,0 xp m,n≥0

=



p, q≥0



m≥0

  um,n+1 x y wp,q xp y q ,

p≥0

m n

M ≥0, N ≥1 m+p=M n+q=N

from which it follows that 

Lw ◦ Lu r ≤

r2M +N

M ≥0, N ≥1

=

≤

1 r

 M ≥0, N ≥1





um,n+1 wp,q



m+p=M n+q=N

 2m+n+1   r

um,n+1 r2p+q wp,q

m+p=M n+q=N

   1  2m+n+1 r

um,n+1

r2p+q wp,q

r m≥0 n≥0

=

p≥0 q≥0

u r w r < ∞. r

It now follows that Lw ◦ L ∈ L(Fr , Er ) and Lw ◦ L ≤ w r /r. Since Lw ◦ Lu (x, 0) = 0 ∈ X, Lw ◦ L ∈ L(Fr , Fr ) and part (a) is proven. (b) It is immediate from the definitions that Lw0 ◦ L is the identity operator on Fr . P ROPOSITION 4.5.2 Suppose that F maps a neighbourhood of the origin in X analytically into X with F (0) = 0 and dF [0] = I. Then there exist open neighbourhoods V, U of the origin in X and an F-analytic function G : V → X such that the following statements are equivalent: F (y) = x, y ∈ U and G(x) = y, x ∈ V. Proof. Suppose that F is analytic from a neighbourhood of 0 ∈ X into X with F (0) = 0 and dF [0] = I. For r > 0 sufficiently small let v, w ∈ Er be defined for (x, y) ∈ Br by v(x, y) = F (y) − x and w(x, y) = v(x, y) − w0 (x, y) = F (y) − x − y. Then w(x, y) = −x +

 dn F [0] yn , n!

n≥2

MULTILINEAR AND ANALYTIC OPERATORS

55

and

w r ≤ r2 +

 dn F [0]

rn ≤ r2 C(F ), n!

n≥2

where C(F ) is a constant determined by F . From the definitions Lv ◦ L − I = (Lv − Lw0 ) ◦ L = Lw ◦ L, and hence, by the preceding lemma, Lv ◦ L − I ≤ rC(F ) for r > 0 sufficiently small. Therefore, for r > 0 sufficiently small, Lv ◦ L is a homeomorphism on Fr by Lemma 2.5.1. So there exists u0 ∈ Fr such that Lv ◦ Lu0 = w0 and, for all (x, y) ∈ Br , Lv ◦ Lu0 (x, y) = ∂y (Lu0 )[(x, y)]v(x, y) − ∂y (Lu0 )[(x, 0)]v(x, 0) = y. In particular, when y ∈ X and t ∈ (0, 1] is sufficiently small,   ty = Lv ◦ Lu0 (0, ty) = ∂y (Lu0 )[(0, ty)] F (ty) − F (0) , which (dividing by t and letting t → 0) gives that ∂y (Lu0 )[(0, 0)] = I, the identity on X. Hence there exists with 0 < < r such that if (x, y) ∈ B , then ∂y (Lu0 )[(x, y)] is a bijection on X. Moreover, for (x, y) ∈ B ,   ∂y (Lu0 )[(x, y)] F (y) − x = y − G(x), (4.21) where G is defined on B2 (X) by G(x) = −∂y (Lu0 )[(x, 0)]v(x, 0) = ∂y (Lu0 )[(x, 0)]x. It is clear that G is F-analytic. Let V = B2 (X) ∩ G−1 (B (X)) and U = B (X) ∩ F −1 (B2 (X)), open neighbourhoods of 0 ∈ X. Since ∂y (Lu0 )[(x, y)] in (4.21) is a bijection, this completes the proof. This result has four important corollaries. T HEOREM 4.5.3 (Analytic Inverse Function Theorem) Suppose that X, Y are Banach spaces, that x0 ∈ U ⊂ X, where U is open. Suppose also that F : U → Y is analytic and that dF [x0 ] ∈ L(X, Y ) is a homeomorphism. Then there exist opens sets U0 and V0 with x0 ∈ U0 ⊂ U , F (x0 ) ∈ V0 ⊂ Y and an analytic map G : V0 → X such that for x ∈ U0 , F (x) ∈ V0 and G(F (x)) = x, and for y ∈ V0 , G(y) ∈ U0 and F (G(y)) = y.

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˜ = U − x0 , replace F : U → Y with F˜ : U ˜ → X defined by Proof. Let U  −1   F˜ (y) = dF [x0 ] F (y + x0 ) − F (x0 ) , and apply the preceding proposition. T HEOREM 4.5.4 (Analytic Implicit Function Theorem) Let X, Y and Z be Banach spaces and suppose that U ⊂ X ×Y is open. Let (x0 , y0 ) ∈ U and suppose that F : U → Z is analytic where the partial derivative ∂x F [(x0 , y0 )] ∈ L(X, Z) is a homeomorphism. Then there exists an open neighbourhood V ⊂ Y of y0 , an open set W ⊂ U and an F-analytic mapping φ : V → X such that (x0 , y0 ) ∈ W and F −1 (z0 ) ∩ W = {(φ(y), y) : y ∈ V }. Proof. In the light of Theorem 4.5.3 the proof is the same as that of Theorem 3.5.4. D EFINITION 4.5.5 A set M ⊂ Fn is called an m-dimensional F-analytic manifold if, for all points a ∈ M , there is an open neighbourhood Ua of 0 ∈ Fm and an analytic function f : Ua → M such that f (0) = a, df [0] is a finite-dimensional linear transformation of rank m, and f maps open sets of U onto relatively open sets of M . R EMARK 4.5.6 Suppose that F : Fn × Fm → Fn is an F-analytic mapping with F (x0 , y0 ) = z0 and that ∂x F [(x0 , y0 )] is a bijection on Fn . Then the analytic implicit function theorem 4.5.4 defines an F-analytic manifold of dimension m by the equation F (x, y) = z0 for (x, y) in a neighbourhood of (x0 , y0 ). T HEOREM 4.5.7 (Composition of Analytic Functions) Suppose X, Y and Z are Banach spaces and that U ⊂ X and V ⊂ Y are open. Suppose that F : U → V and G : V → Z are F-analytic. Then G ◦ F : U → Z is F-analytic. Proof. Let W = U × V × Z and define an analytic function H : W → Y × Z by   H(x, y, z) = F (x) − y, G(y) − z . Let x0 ∈ U , y0 = F (x0 ) ∈ V and z0 = G(y0 ). Then H(x0 , y0 , z0 ) = (0, 0) ∈ Y × Z and y , zˆ) ∂(y,z) H[(x0 , y0 , z0 )](y, z) = (−y, dG[y0 ]y − z) = (ˆ if and only if y. y = −ˆ y and z = −ˆ z − dG[y0 ]ˆ Thus ∂(y,z) H[(x0 , y0 , z0 )] is a homeomorphism. By the analytic implicit function theorem 4.5.4 the solution set, in a neighbourhood of (x0 , y0 , z0 ), of the equation   denotes H(x, y, z) = (0, 0) is described by (y, z) = (Y (x), Z(x)), where (Y , Z)

MULTILINEAR AND ANALYTIC OPERATORS

57

a Y × Z-valued, F-analytic function defined on a neighbourhood of x0 in X. But H(x, y, z) = 0, (x, y, z) ∈ W implies that G(F (x)) = z. Hence G(F (x)) =  Z(x) for x in a neighbourhood of x0 ∈ X. Since x0 ∈ U was chosen arbitrarily, it is immediate that G ◦ F is analytic. The following is the F-analytic version of Proposition 3.6.1 (in the same notation). P ROPOSITION 4.5.8 (Analytic Perturbation of a Simple Eigenvalue) Let X and Y be Banach spaces with X continuously embedded in Y and let s → L(s) be F-analytic from (−1, 1) into L(X, Y ). Suppose also that µ0 is a simple eigenvalue of L(0) with normalized eigenvector ξ0 . Then there exists > 0 and an F-analytic curve {(µ(s), ξ(s)) : s ∈ (− , )} ⊂ R × X such that (µ(0), ξ(0)) = (µ0 , ξ0 ), L(s)ξ(s) = µ(s)ι ξ(s) and ξ(s) = ξ0 + η(s), where ιη(s) ∈ range (L(0) − µ0 ι). Moreover, µ(s) is a simple eigenvalue of L(s) and if |s| < and µ is an eigenvalue of L(s) with |µ0 − µ| < then µ = µ(s). Proof. The proof is identical to that of Proposition 3.6.1, using the analytic implicit function theorem 4.5.4.

4.6 NOTES ON SOURCES The theory of analytic functions is covered in books by Dieudonn´e [28], Federer [29] and Narasimhan [46]. The proofs of the inverse and implicit function theorems, and the demonstration that the composition of analytic functions is analytic, are based on an approach to the Weierstrass preparation theorem in Narasimhan [46]. The treatment of the Fa`a de Bruno formula is taken from Fraenkel [30].

Chapter Five Analytic Functions on Fn We now present some fundamental results about F-analytic functions in finite dimensions.

5.1 PRELIMINARIES Unless otherwise stated F denotes R or C. Consider Fn as an F-linear space the points of which are described by coordinates using the standard basis of real vectors of the form (0, · · · , 1, · · · 0) ∈ Fn . For x = (x1 , · · · , xn ) ∈ Fn and p = (p1 , · · · , pn ) ∈ Nn0 let |x|2 =

n 

|xj |2 ,

j=1

|p| =

n 

pj , xp = xp11 · · · xpnn , p! = p1 !p2 ! · · · pn !

j=1

and ∂f |p| ∂pf = . p 1 ∂xp ∂x1 ∂xp22 · · · ∂xpnn For n ∈ N, let U ⊂ Fn be an open neighbourhood of 0 ∈ Fn and f : U → F an F-analytic function. Then an induction argument starting with the case n = 2 in (4.19) and (4.20) leads to  f (x) = fp xp p∈Nn 0

where fp =

p! 1 ∂pf |fp |r|p| < ∞ (0) ∈ F and sup n p! ∂xp |p|! p∈N0

for some r > 0. This in turn leads to  r|p| |fp | < ∞ for some r > 0. p∈Nn 0

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 n A function so defined is analytic on the open neighbourhood Br (F) of 0 in Fn . (Here Br (F) is the open disc with radius r centred at 0 in F.) D EFINITION 5.1.1 If U ⊂ Cn is open and f : U → C is C-analytic with the property that f (x) ∈ R for all x ∈ U ∩ Rn we say that f is real-on-real. Equivalently f is real-on-real if and only if for all x0 ∈ U ∩ Rn all the terms fp in the expansion of f at x0 are real-on-real multilinear forms on Cn . We use “real-on-real”, instead of “real”, to emphasis the complex setting. R EMARK 5.1.2 Suppose that f : Cn → C is real-on-real. If the basis of Cn is changed to another real basis (a basis of vectors each of which has real coordinates with respect to the standard basis) and points of Cn are now described by coordinates (ζ1 , · · · , ζn ) with respect to that basis, then the function f is a real-on-real function of (ζ1 , · · · , ζn ) ∈ Cn . Therefore a real-on-real C-analytic function remains real-on-real after a real coordinate change of the independent variables. Now we introduce linear spaces of F-analytic functions as follows. For q ∈ N  n−1 and r > 0, let Brq = Brq+1 (F) × Br (F), an open neighbourhood of 0 in Fn , and let Crq denote the space of F-valued F-analytic functions u on Brq with u(0) = 0 of the form  u(x) = up xp p∈Nn 0 , p =0

where  p∈Nn 0,

|up |r(q+1)|p|−qpn = u r,q < ∞.

p =0

  P ROPOSITION 5.1.3 Crq , · r,q is a Banach space. In fact it is a Banach algebra since it is closed under multiplication and uv r,q ≤ u r,q v r,q . Proof. Suppose that {uk }k∈N ⊂ Crq is a Cauchy sequence. Then the corresponding sequence {ukp }k∈N , p = 0, of Taylor coefficients of u at 0 is Cauchy in F. Let ukp → up in F. Since Cauchy sequences are bounded there exists a constant M such that for all P, k ∈ N,  |ukp |r(q+1)|p|−qpn ≤ M. p∈Nn 0 , 0 0, g(x1 , · · · , xn ) = h(x1 , · · · , xn )f (x1 , · · · , xn ) +

q−1  k=0

hk (x1 , · · · , xn−1 )xkn

ANALYTIC FUNCTIONS ON FN

65

for all (x1 , · · · , xn ) ∈ U0 = Brq , where h is analytic on U0 and hk is analytic on n−1  . The functions hk and h are uniquely determined by f and g. V = Brq+1 (F) If Fn = Cn and f and g are real-on-real, then hk and h are real-on-real. Proof. Without loss of generality suppose that v(0) = 1. Choose rˆ such that f, g ∈ Crq for all r with 0 < r ≤ rˆ, where q is given by the hypothesis on f . We will prove the theorem by showing that the linear operator Γ : Crq → Crq defined, in the notation of Lemma 5.1.4, by Γu(x) = f (x)Lu(x) + Au(x), x ∈ Brq , is a bijection on Crq provided r > 0 is sufficiently small. Let v0 (x) = xqn , x ∈ U . Then v0 r,q = rq , f (0, · · · , 0, xn ) = v(xn )v0 (0, · · · , xn ) and hence f − vv0 = Bf, where v is in the statement of the theorem. Therefore by Proposition 5.1.3 and Lemma 5.1.4,

f − v0 r,q ≤ f − vv0 r,q + v0 (1 − v) r,q ≤ Bf r,q + v0 r,q 1 − v r,q ≤ C(f )r1+q + rq 1 − v r,q . From the definition of L, (Γ − I)u = (f − v0 )Lu for u ∈ Crq , and hence

(Γ − I)u r,q = (f − v0 )Lu r,q ≤ Lu r,q (f − v0 ) r,q   ≤ r−q u r,q C(f )r1+q + rq 1 − v r,q . Since v(0) = 1, 1 − v r,q → 0 as r → 0, and it follows that Γ − I ∈ L(Crq , Crq ) with norm less that 1 if r > 0 is sufficiently small. Hence Γ is a bijection on Crq . Hence for g ∈ Crq there is a unique u ∈ Crq with Γu = g. The uniqueness of h and hk now follow from the definition of L and A. If F = C and f and g are real-on-real, then the theorem restricted to Rn ∩ U yields R-analytic functions h and hk . By uniqueness when F = C it follows that h and hk are real-on-real. The proof is complete. R EMARK 5.2.2 The division theorem can be interpreted as saying that f divides g with a remainder that is a polynomial of degree at most (q − 1) in xn , with coefficients that are analytic functions of (x1 , · · · , xn−1 ).

5.3 WEIERSTRASS PREPARATION THEOREM The next result is a special case of the division theorem.

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T HEOREM 5.3.1 Suppose 0 ∈ U (open) ⊂ Fn , f : U → F is analytic, f (0) = 0 and, for (0, · · · , 0, xn ) ∈ U , f (0, · · · , 0, xn ) = xqn v(xn ) where v(0) = 0 and q ≥ 1. Then for r > 0 sufficiently small, h(x1 , · · · , xn )f (x1 , · · · , xn ) = xqn +

q−1 

ak (x1 , · · · , xn−1 )xkn ,

k=0

for all (x1 , · · · , xn ) ∈ U0 = Brq , where h(0) = 0, h and 1/h are analytic on U0 ,  n−1 with ak (0) = 0. The functions ak and and ak is analytic on V = Brq+1 (F) h are uniquely determined by f . If U ⊂ Cn and f is real-on-real, then h and ak are real-on-real. Proof. In the Weierstrass division theorem 5.2.1, choose g(x) = xqn . Then xqn

−

q−1 

hk (x1 , · · · , xn−1 )xkn = h(x1 , · · · , xn )f (x1 , · · · , xn )

k=0

in the neighbourhood U0 of 0 ∈ Fn . In particular, xqn −

q−1 

hk (0, · · · , 0)xkn = h(0, · · · , 0, xn )v(xn )xqn .

k=0

Thus hk (0, · · · , 0) = 0, 0 ≤ k ≤ q−1 and h(0, · · · , 0) = 0 , since v(0) = 0. Since the composition of analytic functions is analytic, 1/h is analytic in a neighbourhood of 0. Let ak = −hk to complete the proof. R EMARK 5.3.2 The preparation theorem has a corollary that in a neighbourhood of 0 ∈ Fn the solution set of f (x1 , · · · , xn ) = 0 coincides with the zero set of a polynomial in xn (the coefficients of which are analytic functions on V ) of the form xqn +

q−1 

ak (x1 , · · · , xn−1 )xkn where ak (0, · · · , 0) = 0.

(5.1)

k=0

The question of how the roots xn depend on (x1 , · · · , xn−1 ) is the basis of the theory to follow.

5.4 RIEMANN EXTENSION THEOREM We begin our study of the level sets of F-analytic functions with a general observation on metric spaces.

ANALYTIC FUNCTIONS ON FN

67

L EMMA 5.4.1 (a) Suppose that a metric space U has a subset G1 that is dense and connected. If G1 ⊂ G ⊂ U , then G is connected. (b) Suppose that G is a closed subset of a non-empty connected metric space U such that U \ G is dense in U . Suppose also that each x ∈ G has an open neighbourhood Vx such that Vx \ G is connected. Then U \ G is connected. Proof. (a) Suppose that G is not connected. Then there exist non-empty, closed subsets F1 and F2 of U such that G ⊂ F1 ∪ F2 , F1 ∩ F2 ∩ G = ∅ and G ∩ Fi = ∅, i = 1, 2. Since G1 is connected, G1 is a subset of one of them, say G1 ⊂ F1 . Hence U = G1 ⊂ F1 , which is a contradiction. This proves (a). (b) Suppose that U \ G is not connected. Then there exist non-empty, open, disjoint sets O1 and O2 whose union is U \ G. Let x ∈ G. Since Vx \ G is connected, it is a subset of O1 or of O2 , but not of both. Let Gi = {x ∈ G : Vx \ G ⊂ Oi }, i ∈ {1, 2}, so that G = G1 ∪ G2 . Suppose that xk ∈ Gi (i fixed) and xk → x ∈ Gj , j ∈ {1, 2} (recall that G is closed). Then Vxk \ G is a subset of O1 or of O2 . Since U \ G is dense, (Vx \ G) ∩ (Vxk \ G) = ∅ for all k sufficiently large. Hence Vxk \ G ⊂ Oj for k sufficiently large. Therefore j = i and Gi is closed, i ∈ {1, 2}. Now let Ui = Oi ∪ Gi . Clearly Ui , i ∈ {1, 2} are non-empty disjoint subsets of U whose union is U . Suppose that {xk } is a convergent sequence in Ui , with limit x. If, for infinitely many k, xk ∈ Gi , then x ∈ Gi ⊂ Ui . Suppose instead that xk ∈ Oi and x ∈ / Oi . Then x ∈ G and xk ∈ Vx \ G for all k sufficiently large. Therefore Vx \ G ⊂ Oi . Hence x ∈ Gi . This shows that Ui is closed in U , i ∈ {1, 2}, and hence U is not connected, a contradiction that proves (b). P ROPOSITION 5.4.2 Suppose that U ⊂ Fn is open, connected and gk : U → F is F-analytic, 1 ≤ k ≤ m. Let E = {x ∈ U : gk (x) = 0 ∈ F, 1 ≤ k ≤ m}. (a) If E = U , then U \ E is dense in U . (b) If, in addition, F = C, then U \ E is connected. Proof. (a) If U \ E is not dense then E contains an open subset of U on which all the functions gk are zero. Hence they are all identically zero on U , by Lemma 4.3.9. Since E = U , this is a contradiction which proves (a). (b) First we observe from Lemma 5.4.1(a) and part (a) above that it suffices to treat the case m = 1. Let E = {x ∈ U : g(x) = 0} where U ⊂ Cn is open and g : U → C is C-analytic. By Lemma 5.4.1 (b) it will suffice, for every x ∈ E, to find an open neighbourhood Vx ⊂ U such that Vx \ E is connected. Without loss of generality suppose that x = 0 ∈ U and that g(0) = 0. Suppose moreover that g ≡ 0 on U . Then we can choose the coordinates (x1 , · · · , xn ) such that g(0, · · · , 0, xn ) ≡ 0. Since xn → g(0, · · · , 0, xn ) is a C-analytic function on a neighbourhood of 0 ∈ C, its zeros are isolated and therefore there exists > 0 such that g(0, · · · , 0, xn ) = 0 if 0 < |xn | ≤ . Hence, by continuity, there exists

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δ > 0 such that g(x1 , · · · , xn ) = 0 if

n−1 k=1

|xk | < δ and 12 < |xn | < . Let

n−1    V0 = (x1 , · · · , xn ) : |xk | < δ, |xn | < . k=1

Now in C (but not in R ) the set n

n

n−1    |xk | < δ, 12 < |xn | <

V˜0 = (x1 , · · · , xn ) : k=1

is a path-connected subset of V0 on which g is nowhere zero. Moreover, for n−1 each fixed (ˆ x1 , · · · , x ˆn−1 ) with k=1 |ˆ xk | < δ, the analytic function xn → ˆn−1 , xn ) has at most a finite number of zeros with |xn | ≤ 3 /4 and g(ˆ x1 , · · · , x the set {x ∈ U : x = (ˆ x1 , · · · , x ˆn−1 , xn ), |xn | ≤ 3 /4, g(x) = 0} ⊂ Cn is path-connected. Since V˜0 is path-connected, V0 \ E is path-connected, and hence connected. Since V0 is an open neighbourhood of 0, where 0 represents an arbitrary point of E, this completes the proof. The following is a particular case of a classical theorem which holds more generally. T HEOREM 5.4.3 (Riemann Extension Theorem) Suppose that U ⊂ Cn is open and gk : U → C is C-analytic, 1 ≤ k ≤ m. Let E = {x ∈ U : gk (x) = 0 for all k, 1 ≤ k ≤ m} and suppose that f is C-analytic on U \ E with sup{|f (x)| : x ∈ U \ E} < ∞. Then there exists a function f˜ which is C-analytic on U and f = f˜ on U \ E. Proof. Since   U \ E = ∪m k=1 U \ Ek where Ek = {x ∈ U : gk (x) = 0} it suffices to prove the required result when m = 1. Since analyticity is defined locally it will suffice to choose x ∈ E and to show that the result is true when U is replaced by some open neighbourhood of x. Without loss of generality suppose that 0 ∈ E is the point in question. Let , δ > 0 and V0 , a neighbourhood of 0 ∈ U ∩ E, be as defined in theproof of part (b) of the last proposition. Then for n−1 any fixed (x1 , · · · , xn−1 ) with k=0 |xk | < δ, the set {z ∈ C : |z| ≤ 3 /4 and (x1 , · · · , xn−1 , z) ∈ E} ⊂ {z ∈ C : |z| ≤ 3 /4 and g1 (x1 , · · · , xn−1 , z) = 0}

ANALYTIC FUNCTIONS ON FN

69

is finite and therefore z → f (x1 , · · · , xn−1 , z) is C-analytic except at finitely many points and bounded on {z ∈ C : |z| ≤ 3 /4}. Note also that z → f (x1 , · · · , xn−1 , z) is analytic in a neighbourhood of the circle |z| = 3 /4 for any such fixed (x1 , · · · , xn−1 ), since then (x1 , · · · , xn−1 , z) ∈ V˜0 and g does not vanish on V˜0 . Therefore the singularities of this function of z must be removable and, by Cauchy’s integral formula, the function
π f (x1 , · · · , xn−1 , 3 eit /4) 3 eit  1 ˜ f (x1 , · · · , xn ) = dt (5.2) 2π −π 3 eit /4 − xn 4 extends f to all of V0 . For fixed (x1 , · · · , xn−1 ) as above, xn → f˜(x1 , · · · , xn ) is C-analytic. However it is clear, from the definition of f˜ given in (5.2) that f˜ is Canalytic on V0 ⊂ Cn . This completes the proof. The function f˜ is called an analytic extension of f .

5.5 NOTES ON SOURCES These classical results are in the books by Chow and Hale [19], Dieudonn´e [28], Federer [29], Golubitsky and Guillemin [34]. For results on functions of a complex variable, see Ahlfors [1], Cartan [15], Chatterji [17].

Chapter Six Polynomials

6.1 CONSTANT COEFFICIENTS A polynomial of the form A(Z) = ap Z p + . . . + a1 Z + a0 , p ∈ N0 , ap = 0, with complex coefficients is said to have degree p and a complex number z such that A(z) = 0 is called a root of A(Z). The fundamental theorem of algebra says that A(Z) has at most p roots, z1 , · · · , zk , say, and that A(Z) = ap (Z − z1 )m1 · · · (Z − zk )mk , where m1 + · · · + mk = p

(6.1)

and the zj s are distinct. In this factorization of A(Z) over C, the number mj is called the multiplicity of the root zj . If mj = 1 then zj is called a simple root of A(Z), otherwise it is a multiple root. The coefficient of Z p is called the principal coefficient of A(Z). Continuous Dependence of Roots P ROPOSITION 6.1.1

Let p ≥ 1. (a) If z ∈ C is a simple root of a polynomial ap−1 Z p−1 + · · · +  a0 Zp + 

with complex coefficients, then there is a C-analytic function f , defined in a neighap−1 ), such that z = f (a0 , · · · , ap−1 ) is a simple root of the bourhood of ( a0 , · · · ,  polynomial Z p + ap−1 Z p−1 + · · · + a0 and z = f ( a0 , · · · ,  ap−1 ). If, in addition,  a0 , · · · ,  ap−1 , z, a0 , · · · , ap−1 ∈ R, then f (a0 , · · · , ap−1 ) ∈ R. (b) Suppose that z ∈ C is a root of multiplicity q ≥ 1 of the polynomial in part (a), and the distance of all the other roots from z is at least 

> 0. Then, for all

with 0 < < 

there exists δ > 0 such that the polynomial Z p + ap−1 Z p−1 + . . . + a0 has exactly q complex roots, counted according to their multiplicities, in the set {z ∈ C : |z − z| < } a0 |, . . . , |ap−1 −  ap−1 | < δ. provided that |a0 −  (c) For all > 0, there exist δ > 0 such that |z| < for all z ∈ C with z p + ap−1 z p−1 + · · · + a0 = 0 when |a0 |, · · · , |ap−1 | < δ.

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Similarly, if |a0 | + · · · |ap−1 | ≤ M then |z| ≤ m(M ), where m(M ) depends only on M . Proof. (a) Since z is a simple root of the polynomial,   d p p−1 ap−1 Z + ... + a0 ) = 0. (Z +  dZ z Z= Therefore the analytic implicit function theorem 4.5.4 ensures the existence of a a0 , . . . ,  ap−1 ) in C-analytic function f such that, for all (a0 , . . . , ap−1 ) close to ( Cp ,  {Z p + ap−1 Z p−1 + · · · + a0 }Z=f (a0 ,··· ,ap−1 ) ≡ 0 and

  d (Z p + ap−1 Z p−1 + . . . + a0 ) = 0. dZ Z=f (a0 ,... ,ap−1 )

The same reasoning in the case of a polynomial with real coefficients completes the proof of the first part. (b) Part (b) is immediate from Rouch´e’s theorem. (c) If part (c) is false, then there exist sequences {a0,n }, · · · , {ap−1,n }, {zn } ⊂ C such that lim aj,n = 0 for 0 ≤ j ≤ p − 1, lim |zn | ∈ (0, ∞]

n→∞

n→∞

and, for all n ∈ N, znp + ap−1,n znp−1 + · · · + a0,n = 0. Thus we obtain the contradiction that ap−2,n a0,n − · · · − p−1 → 0, zn = −ap−1,n − zn zn and the proof is complete. Greatest Common Divisors Consider two polynomials of the complex variable Z, with constant coefficients in C, given by A(Z) = ap Z p + · · · + a1 Z + a0 , p ≥ 1, ap = 0, B(Z) = bq Z q + · · · + b1 Z + b0 , q ≥ 1, bq = 0. We will say that their greatest common divisor is the polynomial with largest degree m ∈ N0 of the form Z m + cm−1 Z m−1 + · · · + c0 , with coefficients cj ∈ C, which

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divides both A(Z) and B(Z). Then A(Z) and B(Z) are said to be co-prime if the constant polynomial 1 of degree 0 is their greatest common divisor. An elementary criterion says that A(Z) and B(Z) are not co-prime if and only if there exist two polynomials P (Z) and Q(Z) such that A(Z)P (Z) + B(Z)Q(Z) = 0, P (Z) ≡ 0, Q(Z) ≡ 0, deg(P (Z)) < q, deg(Q(Z)) < p. Equivalently, if P (Z) = cq−1 Z q−1 + · · · + c1 Z + c0 and Q(Z) = dp−1 Z p−1 + · · · + d1 Z + d0 , p ≥ q, the equation Ax = 0 has a solution x = (c0 , · · · , cq−1 , d0 , · · · , dp−1 )T = 0 if and only if A(Z) and B(Z) are not co-prime, where the square matrix A is given by  0 ≤ i − j ≤ p, 1 ≤ j ≤ q  ai−j , bq+i−j , 0 ≤ j − i ≤ q, q + 1 ≤ j ≤ p + q . Aij =  0, otherwise The complex (p + q) × (p + q) matrix A is called the resultant matrix of A(Z) and B(Z). Its determinant, R(a0 , · · · , ap ; b0 , . . . , bq ), is called the resultant of A(Z) and B(Z). The resultant is a polynomial in the coefficients a0 , · · · , ap , b0 , · · · , bq which is zero if and only if A(Z) and B(Z) are not co-prime. L EMMA 6.1.2 Let A(Z) be a polynomial of degree p ≥ 1 and let A
(Z) denote its derivative. Denote by C(Z) the greatest common divisor of A(Z) and A
(Z) and let A(Z) = C(Z)E(Z) for some polynomial E(Z). Then E(Z) and E
(Z) are co-prime and E(Z) has the same roots as A(Z).

Proof. Suppose A(Z) is given by (6.1). Then A
(Z) = P (Z)C(Z) where

C(Z) =

m ˆ 

ˆ 1 −1 ˆk ˆ −1 (Z − zˆ1 )m · · · (Z − zˆkˆ )m

ˆ k=1

and {ˆ zkˆ : kˆ = 1, · · · m}, ˆ zˆkˆ of multiplicity m ˆ kˆ > 1, denotes the set of multiple roots of A(Z), and no root of A(Z) is a root of P (Z). Hence C(Z) is the greatest common divisor of A(Z) and A
(Z), and A(Z) = C(Z)E(Z) where all the roots of E(Z) are simple and the roots of E(Z) coincide with those of A(Z).

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For future reference, note that when p = q the matrix A has the form



a0 a1 .. .

      ap−1   ap   0   .  .. 0

0 a0 .. .

... ...

0 0 .. .

b0 b1 .. .

0 b0 .. .

... ...

0 0 .. .

ap−2 ap−1 ap .. .

... ... ...

a0 a1 a2 .. .

bp−1 bp 0 .. .

bp−2 bp−1 bp .. .

... ... ...

b0 b1 b2 .. .

0

...

ap

0

0

...

bp

             

.

(6.2)

Discriminant of a Polynomial Consider the polynomial A(Z) = ap Z p + . . . + a1 Z + a0 , p ≥ 1, ap = 0, the coefficients of which are complex. It is clear from (6.1) that zj ∈ C is a multiple root of A(Z) if and only if (Z − zj ) is a common factor of A(Z) and A
(Z), where A
(Z) denotes the polynomial obtained by differentiating A with respect to Z. Indeed A(Z) has no multiple roots if and only if A(Z) and A
(Z) are co-prime. For (a0 , · · · , ap ) ∈ Cp+1 define D(a0 , . . . , ap ) ∈ C to be the resultant of A(Z) and A
(Z). This is called the discriminant of A(Z). Notice that D is a polynomial in the p + 1 variables a0 , . . . , ap which vanishes exactly when A(Z) has at least one multiple root. E XAMPLE 6.1.3 (Quadratic polynomials) p = 2, A(Z) = a2 Z 2 + a1 Z + a0 , A
(Z) = 2a2 Z + a1 = B(Z) and, with a2 = 0,    a0 b 0 0    D(a0 , a1 , a2 ) = R(a0 , a1 , a2 ; b0 , b1 ) =  a1 b1 b0   a2 0 b1     a0 a1 0    =  a1 2a2 a1  = −a2 (a21 − 4a0 a2 ),  a2 0 2a2  which vanishes exactly when the usual discriminant a21 − 4a0 a2 = 0.

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6.2 VARIABLE COEFFICIENTS In the Weierstrass preparation theorem 5.3.1, we saw polynomials in xn of the form xqn +

q−1 

ak (x1 , · · · , xn−1 )xkn ,

k=0

in which the coefficients ak are F-analytic functions of n − 1 variables. In this section we consider such polynomials in the special case when F = C. Let V ⊂ Cm , m ≥ 1 be given by V = {(z1 , · · · , zm ) ∈ Cm : |zj | < δ, 1 ≤ j ≤ m}, for some δ > 0 and consider polynomials of the form ap (z1 , · · · , zm )Z p + · · · + a1 (z1 , · · · , zm )Z + a0 (z1 , · · · , zm ), where the coefficients ak : V → C are C-analytic functions and ap ≡ 0. (The case m = 0 will correspond to polynomials in Z with constant coefficients.) This polynomial, which we denote by A(Z; z1 , · · · , zm ), has degree p. The coefficients of the polynomial B(Z; z1 , · · · , zm ) are functions bk : V → C, where it is understood that A(Z; z1 , · · · , zm ) and B(Z; z1 , · · · , zm ) may have different degrees. (When the meaning is clear we refer simply to polynomials A or B on V .) We say that a polynomial is real-on-real if and only if all its coefficients are real-on-real analytic functions. The discriminant D = D(a0 , · · · , ap ) of the polynomial A is an analytic function defined on V ⊂ Cm by D(a0 , · · · , ap )(ξ) = D(a0 (ξ), · · · , ap (ξ)), ξ = (z1 , · · · , zm ) ∈ V.

(6.3)

Greatest Common Divisors From the previous viewpoint, {A(Z; z1 , · · · , zm ) : (z1 , · · · , zm ) ∈ V } is a family of polynomials parameterized by (z1 , · · · , zm ) ∈ V . The notion of the greatest common divisor of two polynomials A and B is therefore more subtle in this case. D EFINITION 6.2.1 Let W ⊂ V . The greatest common divisor of A and B on W is a polynomial C(Z; z1 , · · · , zm ) of degree d, say, where the coefficients ck are C-analytic on V , cd ≡ 1 on V and, for every (z1 , · · · , zm ) ∈ W , the polynomial C(Z; z1 , · · · , zm ) is the greatest common divisor of A(Z; z1 , · · · , zm ) and B(Z; z1 , · · · , zm ) in the sense of §6.1. The first question is whether in this general setting two polynomials have a greatest common divisor. T HEOREM 6.2.2

(Euclid’s Algorithm) Let polynomials A(Z; z1 , · · · , zm ) and B(Z; z1 , · · · , zm )

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POLYNOMIALS

have degree p and q, respectively, where at least one of ap and bq is identically equal to 1 on V . Then there exists a polynomial C(Z; z1 , · · · , zm ) of degree r on V and a C-analytic function g, such that C is the greatest common divisor of A and B on the set W = V \ G, where G = {(z1 , · · · , zm ) : g(z1 , · · · , zm ) = 0} =  V . Note that W is a dense connected subset of V , by Proposition 5.4.2. Suppose that ap ≡ 1. Then there is a polynomial E(Z; z1 , · · · , zm ) such that A = CE for all (Z; z1 , · · · , zm ) ∈ C × V . If r = 0 then C ≡ 1 on V . Proof. Let P1 = A, m1 = p and P2 = B, m2 = q, if p ≥ q, P2 = A, m2 = p and P1 = B, m1 = q, if p < q. Suppose for j ∈ N that Pk , 1 ≤ k ≤ j + 1, are given polynomials of degree mk such that the coefficient of Z mk is gk ≡ 0 and that mk is a non-increasing function of k. Now let Q(Z; z1 , · · · , zm ) = Z mj −mj+1 gj (z1 , · · · , zm )Pj+1 (Z; z1 , · · · , zm ) − gj+1 (z1 , · · · , zm )Pj (Z; z1 , · · · , zm ). The crucial observation is that, on the set Wj = {(z1 , · · · , zm ) ∈ V : gj+1 (z1 , · · · , zm ) = 0}, the greatest common divisor of Pj and Pj+1 is also the greatest common divisor of Q and Pj+1 . Obviously the degree of Q is smaller than the degree of Pj . If the degree of Q is not larger than that of Pj+1 let Pj+2 = Q; otherwise rename Pj+1 as Pj+2 and replace the old Pj+1 by Q. This defines a set of polynomials P1 , · · · , Pj+2 with mk > mk+2 for all k ≤ j. Clearly this process terminates after a finite number of steps when 0 = gJ+1 PJ − gJ Z mJ −mJ+1 PJ+1 , say, for some J ∈ N. Hence, at every point (z1 , · · · , zm ) ∈ V at which the product g = g1 · · · gJ+1  = PJ+1 /gJ+1 is the of the highest order coefficients of the Pj s is non-zero, C greatest common divisor of P1 and P2 . This defines g and consequently G in the statement of the theorem. By definition, g ≡ 0, and so G = V . Now suppose that ap ≡ 1. Then at every point (z1 , · · · , zm ) ∈ V \ G the roots  of C(Z; z1 , · · · , zm ) form a subset of those of A(Z; z1 , · · · , zm ) and are therefore bounded when (z1 , · · · , zm ) lies in a compact subset of V , by Proposition 6.1.1 (c).  are polynomial functions of the roots of C,  it follows Since the coefficients of C that these coefficients are bounded on subsets of V \ G that are compact in V . Therefore, by the Riemann extension theorem 5.4.3, they have analytic extension  extended to to all of V . We denote by C the polynomial with the coefficients of C  V . To obtain the existence of E, note first that a polynomial E is defined on V \ G  on V \ G. The argument for extending E  as a polynomial on by writing A = C E  V is the same as that for C.

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R EMARKS 6.2.3 If A and B have coefficients that are real-on-real, then Euclid’s algorithm obviously leads to a greatest common divisor C and a polynomial E, both of which are real-on-real. If ap ≡ 1, bq ≡ 1, and R(a0 , · · · , ap−1 , 1; b0 , · · · , bq−1 , 1) ≡ 0 (in which case it is non-zero on a connected open dense set in V ), we may take g = R(a0 , · · · , ap−1 , 1; b0 , · · · , bq−1 , 1), r = 0 and c0 = C ≡ 1. If r ≥ 1, then R(a0 , · · · , ap−1 , 1; b0 , · · · , bq−1 , 1) ≡ 0. T HEOREM 6.2.4 (Simplifying a Polynomial) Let A(Z; z1 , · · · , zm ) be a polynomial of degree p with discriminant D(a0 , · · · , ap−1 , 1) ≡ 0 and ap ≡ 1 on V . Then there exists another polynomial E(Z; z1 , · · · , zm ) of degree q, say, with eq ≡ 1 such that E(Z; z1 , · · · , zm ) has the same roots as A(Z; z1 , · · · , zm ), possibly with smaller multiplicities, and D(e0 , · · · , em−1 , 1) ≡ 0 on V . (In particular, for (z1 , · · · , zm ) in an open dense connected subset W of V , E(Z; z1 , · · · , zm ) has no multiple roots.) If A is real-on-real, then so is E. (E is called the simplification of A.) Proof. For (z1 , · · · , zm ) ∈ V , the polynomial A(Z; z1 , · · · , zm ) has a multiple root if and only if its discriminant is zero. It therefore suffices to let E be the polynomial given by Euclid’s algorithm for which A = CE where C is the greatest common divisor of A and A
on W , an open dense connected subset of V . An appeal to Lemma 6.1.2 completes the proof. T HEOREM 6.2.5 (Projection Lemma) Let A1 (Z; z1 , · · · , zm ) be a polynomial of degree p with ap ≡ 1 on V and let Aj (Z; z1 , · · · , zm ) be a polynomial of degree at most p − 1, 2 ≤ j ≤ k. Let  A = (z1 , · · · , zm ) ∈ V : there exists z ∈ C

 with Aj (z; z1 , · · · , zm ) = 0 for all j ∈ {1, · · · , k} .

Then there exists a finite family {Rα : α ∈ Σ} of analytic functions on V such that   A = (z1 , · · · , zm ) ∈ V : Rα (z1 , · · · , zm ) = 0 for all α ∈ Σ . If the polynomials Aj are real-on-real, then the functions in A are real-on-real. R EMARK 6.2.6 It is worth noting that this result is false in the setting of realanalytic functions. For example let m = 1 and V = (− , ), k = 2, A1 (X, x1 ) = X 2 − x1 and A2 ≡ 0. Then p = 2 and A = [0, ) which is not the zero-set of any real-analytic function defined on (− , ), because of Lemma 4.3.9. Proof. For any t2 , · · · , tk ∈ C, let R(t2 , · · · , tk ; z1 , · · · , zm ) denote the resulk tant of two polynomials, A1 and A1 + j=2 tj Aj , of the same degree. Let the coefficients of Aj be denoted by aji , 1 ≤ j ≤ k, and in (6.2) let ai = a1i and k bi = ai + j=2 tj aji . The resultant R may now be obtained from formula (4.2)

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POLYNOMIALS

for the determinant of the matrix (6.2) with these coefficients. By subtracting the j th column from the p + j th column, the coefficients ai can be eliminated from the right half of the matrix without changing its determinant and the non-zero entries k in the right half of the resulting matrix are all of the form j=2 tj aji . Therefore 

R(t2 , · · · , tk ; z1 , · · · , zm ) =

tα Rα (z1 , · · · , zm )

α∈Nk−1 0 |α|=p

where t = (t2 · · · , tk ) and Rα : V → C is analytic and independent of t. Suppose that (z1 , · · · , zm ) ∈ A. Then, for some z ∈ C, Aj (z, z1 , · · · , zm ) = 0 for all j ∈ {1, · · · , k}. Therefore, for all t2 , · · · , tk , z is a common root of the polynomials A1 and A1 + k m−1 j=2 tj Aj . Therefore R(t2 , · · · , tk ; z1 , · · · , zm ) = 0 for all (t2 , · · · , tk ) ∈ C k−1 and so Rα (z1 , · · · , zm ) = 0 for all α ∈ Σ where Σ = {α ∈ N0 : |α| = p}. Conversely, suppose (z1 , · · · , zm ) ∈ V and Rα (z1 , · · · , zm ) = 0 for α ∈ Σ. It follows that R(t2 , · · · , tk ; z1 , · · · , zm ) = 0 for (t2 , · · · , tk ) ∈ Ck−1 . Therefore k the polynomials A1 and A1 + j=2 tj Aj have a common root (possibly depending on (t2 , · · · tk )) for all (t2 , · · · , tk ) ∈ Ck−1 . Let ζ1 , · · · , ζν , ν ≤ p, be the distinct roots of A1 (Z; z1 , · · · , zm ) and for 1 ≤ i ≤ ν let Yi = {(t2 , · · · , tk ) ∈ Ck−1 :

k 

tj Aj (ζi ; z1 , · · · , zm ) = 0}.

j=2

Each Yi is a linear subspace of Ck−1 and their union is Ck−1 . Hence Yi0 = Ck−1 for some i0 ∈ {1, · · · , ν} and so Aj (ζi0 ; z1 , · · · , zm ) = 0 for all j ∈ {1, · · · , k}. It is clear from the construction that if the polynomials Aj are real-on-real, then the functions Rα are real-on-real. This completes the proof.

6.3 NOTES ON SOURCES The theory of polynomials with constant coefficients used here is extensively treated in Van der Waerden’s classic [65]. The non-constant coefficient theory is to be found in Mumford [45] and Narasimhan [46].

Chapter Seven Analytic Varieties Once again the field F is either C or R. Let a ∈ Fn , n ∈ N. Two subsets S and T of Fn are said to be equivalent at a if there is an open neighbourhood O of a such n that O ∩ S = O ∩ T . We note that this is an equivalence relation on 2F and write S ∼a T . The corresponding equivalence class, denoted by γa (S) for S ⊂ Fn , is called the germ of S at a and if S˜ ∈ γa (S) we say that S˜ is a representative of γa (S). Since {a} ∩ U = {a} and ∅ ∩ U = ∅ for all open sets U containing a, we write γa ({a}) = {a} and γa (∅) = ∅. If a ∈ / S, γa (S) = ∅. The finite unions, intersections and complements of germs of sets at a are defined by the same operations on representatives. (It is easy to check that these are well-defined operations on germs, and independent of the chosen representatives.)

7.1 F-ANALYTIC VARIETIES D EFINITION 7.1.1 Suppose that U ⊂ Fn is a non-empty open set and that G denotes a finite collection of functions g : U → F which are F-analytic on U . Let var (U, G) = {x ∈ U : g(x) = 0 for all g ∈ G}. This is called the F-analytic variety generated by G on U . If U ⊂ Cn and the elements of G are real-on-real, we say that var (U, G) is real-on-real provided that U ∩ Rn = ∅. A point x ∈ var (U, G) is said to be m-regular if there is a neighbourhood O of x in Fn such that O ∩ var (U, G) is an F-analytic manifold of dimension m (see Definition 4.5.5). Note that var (U, G1 ) ∩ var (U, G2 ) = var (U, G1 ∪ G2 ) var (U, G1 ) ∪ var (U, G2 ) = var (U, G3 ) where G3 = {g1 g2 : gi ∈ Gi , i = 1, 2}. The germ at a of an F-analytic variety is referred to as an F-analytic germ and the germ of a real-on-real C-analytic variety in Cn is called a real-on-real germ. The set of all F-analytic germs at a ∈ Fn is denoted by Va (Fn ). If α ∈ Va (Fn ), its dimension, dimF α, is the largest integer m such that every representative of α contains an m-regular point (the point a itself need not be m-regular.) If no such integer exists, we say that dimF α = −1.

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R EMARKS 7.1.2 For a ∈ Fn , Fn , {a} and ∅ are elements of Va (Fn ) with dimF ∅ = −1, dimF {a} = 0 and dimF γa (Fn ) = n. Theorem 6.2.5 says that γ0 (A) ∈ V0 (Cm ). If α, β ∈ Va (Fn ), then both α ∩ β and α ∪ β are in Va (Fn ), but in general α\β ∈ / Va (Fn ) L EMMA 7.1.3 Suppose that M ⊂ Fn is an F-analytic manifold (Definition 4.5.5) and a ∈ M . Then γa (M ) ∈ Va (Fn ). If a ∈ U ∩ M and var (U, G) is an F-analytic variety, there is an open neighbourhood W of a in M such that W \ var (U, G) is either empty or dense in W . Proof. First we show that γa (M ) is in Va (Fn ). Without loss of generality suppose that 0 = a ∈ M and, in the notation of Definition 4.5.5, let Z1 = range df [0], Fn = Z1 ⊕ Z2 , and write f (x) = f1 (x) + f2 (x) ∈ Z1 ⊕ Z2 , x ∈ U0 ⊂ Fm , where U0 is a neighbourhood of 0 ∈ Fm . By hypothesis, df [0] has rank m and df1 [0] : Fm → Z1 is a bijection. By the analytic inverse function theorem 4.5.3, U0 in Definition 4.5.5 can be chosen so that f1 , from U0 ⊂ Fm onto a neighbourhood W0 of f1 (0) = 0 ∈ Z1 , is a bijection with an analytic inverse. Now {f (x) : x ∈ U0 } is a representative of γ0 (M ) in Fn and {f (x) : x ∈ U0 } = {(f1 (x), f2 (x)) : x ∈ U0 } = {(y, f2 ◦ f1−1 (y) : y ∈ W0 } = {(y, z) ∈ W0 × Z2 : z − f2 ◦ f1−1 (y) = 0}. Since f2 ◦ f1−1 is analytic this shows that γ0 (M ) ∈ V0 (Fn ). Now suppose that var (U, G) is an F-analytic variety in Fn , let 0 ∈ M and let U0 as above. Suppose that B ⊂ U0 is a ball centred at 0 ∈ Fm and let W = f (B). Then 0 ∈ W , which is a relatively open connected subset of M . ˆ ⊂W Suppose that W \var (U, G) is not dense in W . Then there is an open set W ˆ ⊂ var (U, G). Let B ˆ = f −1 (W ˆ ). Then g ◦ f ≡ 0 on B ˆ for all g ∈ G. such that W Hence g ◦ f ≡ 0 on B for all g ∈ G, by Theorem 4.3.9. Hence W ⊂ var (U, G), in other words, W \ var (U, G) is empty. This proves the result. L EMMA 7.1.4 Let var (U, G) be an F-analytic variety in Fn and M ⊂ U a connected F-analytic manifold such that M ∩ var (U, G) has non-empty interior relative to M . Then M ⊂ var (U, G). Proof. Let N o denote the relative interior in M of N = M ∩ var (U, G). By definition, N o is open in M , and non-empty by hypothesis. Suppose that x belongs to the boundary in M of N o . By Lemma 7.1.3 there is an open neighbourhood W of x in M such that W \ var (U, G) is either empty or dense in W . Now W ∩ N o = ∅ since x is on the boundary of N o and, since N o ⊂ M ∩ var (U, G) is open, W \ var (U, G) is not dense in W . Hence it is empty, which implies that x ∈ N o . Thus N o is closed in M . By connectedness, N o = M and M ⊂ var (U, G).

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D EFINITION 7.1.5 A germ α ∈ Va (Fn ) is said to be irreducible if α = α1 ∪ α2 for germs α1 , α2 ∈ Va (Fn ) implies that α = α1 or α = α2 . For example, ∅ and {a} and are irreducible elements of Va (Fn ). L EMMA 7.1.6 If M is an m-dimensional F-analytic manifold and a ∈ M , then γa (M ) ∈ Va (Fn ) is irreducible. Proof. By Lemma 7.1.3, γa (M ) ∈ Va (Fn ). To see that it is irreducible suppose that E1 and E2 are F-analytic varieties in Fn such that γa (M ) = γa (E1 ∪ E2 ) and γa (E1 ) = γa (M ) = γa (E2 ). It follows also from Lemma 7.1.3 that for i = 1, 2 there exists an open neighbourhood W of a in M such that (M \ Ei ) ∩ W is either empty or dense in W . Note M and E1 ∪ E2 coincide in a neighbourhood U of a in Fn . Hence if (M \ E1 ) ∩ W is empty, it follows that E2 ⊂ E1 in a neighbourhood of a and that γa (M ) = γa (E1 ), which is assumed to be false. If (M \ E2 ) ∩ W is empty we reach a similar contradiction. Therefore (M \ Ei ) ∩ W is dense, i = 1, 2 and so (M \ (E1 ∪ E2 )) ∩ W is dense in W . But this contradicts the fact that γa (M ) = γa (E1 ∪ E2 ) and proves that γa (M ) is an irreducible germ. We will see another important example of irreducible germs in Lemma 7.2.9. The most elementary non-trivial example of an analytic variety is one which is defined as the zeros in an open set U ⊂ Fn , n ≥ 2, of a single F-analytic function f : U → F. Suppose, without loss of generality, that 0 ∈ U , that f (0) = 0 and that f ≡ 0 on U . Then the Weierstrass preparation theorem 5.3.1 gives that there exists a choice of coordinates (x1 , · · · , xn ) ∈ Fn , r > 0 and an open set V ⊂ Fn−1 containing 0 such that, with U0 = V × Br (F), var (U0 , {f }) = var (U0 , {h}), h(x1 , · · · , xn ) = A(xn ; x1 , · · · , xn−1 ), where A(X; x1 , · · · , xn−1 ) is a polynomial with coefficients that are analytic functions of (x1 , · · · , xn−1 ) ∈ V of the form A(X; x1 , · · · , xn−1 ) = Xq +

q−1 

ak (x1 , · · · , xn−1 )X k , (x1 , · · · , xn−1 ) ∈ V,

k=0

with ak (0) = 0, 0 ≤ k ≤ q − 1. Since real polynomials need not have real roots, we need special structure to take this idea any further when F = R. However when F = C polynomials do have roots in C and we have the following. T HEOREM 7.1.7 When F = C the polynomial A above can be chosen with the following additional properties:

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(a) Its discriminant (see (6.3)) D = D(a0 , · · · , aq−1 , 1) ≡ 0 on V .   (b) Every point of var (U0 , {f })\ var (V, {D})×C is an (n−1)-regular point of var (U0 , {f }). (c) dimC α = n − 1 where α is the germ of var (U0 , {f }). (d) If f is real-on-real, then A is real-on-real. Proof. (a) We simplify the polynomial obtained from the Weierstrass preparation theorem using Theorem 6.2.4. This may change the value of q and the coefficients ak , but we retain the original notation for the simplified polynomial. Since, after simplification, A(Z; z1 , · · · , zn−1 ) has no multiple zeros for (z1 , · · · , zn−1 ) in an open dense connected subset of V (see Proposition 5.4.2), this  proves(a). (b) Since, for every (z1 , · · · , zn ) ∈ var (U0 , {f }) \ var (V, {D}) , Z = zn is a simple zero of A(Z; z1 , · · · , zn−1 ), the analytic implicit function theorem 4.5.4 gives (b). (c) Since ak (0) = 0 and ak is continuous on V , 0 ≤ k ≤ q − 1, we know from Proposition 5.4.2   and Proposition 6.1.1 that there are points of var (U0 , {h}) \ var (V, {D}) × C arbitrarily close to zero. Therefore there are (n − 1)-regular points of var (U0 , {f }) arbitrarily close to 0. Thus dimC α = n − 1. (d) This is guaranteed by the Weierstrass preparation theorem 5.3.1 and Theorem 6.2.4. R EMARK 7.1.8

Of course, γ0 (var (U, {f })) = γ0 (var (U0 , {f })).

Now we will develop the ideas involved in the proof of this result to obtain something much more general.

7.2 WEIERSTRASS ANALYTIC VARIETIES Throughout F = C, m ∈ N0 and, for m ∈ N, V ⊂ Cm is given by V = {(z1 , · · · , zm ) ∈ Cm : |zk | < δ, 1 ≤ k ≤ m}. In the case m = 0, V = {0}. D EFINITION 7.2.1 When m ∈ N, a Weierstrass polynomial on V is a polynomial A(Z; z1 , · · · , zm ), (z1 , · · · , zm ) ∈ V , of the form Zp +

p−1 

ak (z1 , · · · , zm )Z k , p ∈ N,

(7.1)

k=0

where a0 (0) = · · · = ap−1 (0) = 0 and D(a0 , · · · , ap−1 , 1) ≡ 0 on V. (By Proposition 5.4.2, D(a0 , · · · , ap−1 , 1) = 0 on a connected, open, dense subset of V .) When m = 0 Weierstrass polynomials are of the form Z p , p ∈ N.

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R EMARK 7.2.2 Suppose that the coefficients ak in a polynomial of the form (7.1) vanish at 0 ∈ V and the discriminant is identically zero on V . Then its simplification, Theorem 6.2.4, is a Weierstrass polynomial on V . (All its coefficients, apart from the principal coefficient, are zero at 0 ∈ V .) If A, B are Weierstrass polynomials on V and C is any (non-constant in Z) polynomial on V with AC = B, then C is a Weierstrass polynomial. Let n > m ∈ N0 and consider a family {Am+1 , · · · , An } of Weierstrass polynomials on V . For each k ∈ {m + 1, · · · , n} let hk (z1 , · · · , zn ) = Ak (zk ; z1 , · · · , zm ). Let H denote the family of n − m functions hk defined in this way. (Each hk ∈ H is a polynomial in zk with coefficients that are analytic functions on V ⊂ Cm . Thus hk is independent of all zj for j ∈ {m + 1, · · · , n} \ {k}.) D EFINITION 7.2.3 A Weierstrass analytic variety is a set in Cn of the form  n−m , H , 0 ≤ m < n. For our purposes, a Weierstrass analytic variety var V × C is identified with the set H of Weierstrass analytic polynomials which define it and, if m ∈ N, its discriminant D(H) : V → C is the product of the discriminants of the polynomials Ak used in the definition.   For m ∈ N, the branches of a Weierstrass analytic variety var V × Cn−m , H are the components of     var V × Cn−m , H \ var (V, D(H)) × Cn−m .  7.2.4 All points on a branch of a Weierstrass analytic variety var V× R EMARKS  n−m C , H , for n ∈ N and 0 < m < n, are m-regular because, by the analytic implicit function theorem 4.5.4, in a neighbourhood of such a point each of the coordinates zm+1 , · · · , zn depends locally analytically on (z1 , · · · , zm ) ∈ V . Thus each branch is a connected C-analytic manifoldof dimension m and by Proposition 5.4.2 it projects onto the connected set V \ var V, {D(H)} . But it is not possible in general to define any one of the coordinates zm+1 , · · · , zn as an analytic function on V \ var (V, {D(H)}) when the latterset is multiply connected. In the proof  of Corollary 7.4.4 we will see that E = var V × Cn−m , H contains no manifold of dimension strictly greater that m and hence dimC γ0 (E) = m. When m = 0, the only Weierstrass analytic variety in Cn is {0} since all Weierstrass polynomials are of the form Z p , p ∈ N. The following are a few elementary examples. E XAMPLES 7.2.5 In Definition 7.2.1 with m = 1, n = 2 and V = C, the polynomial A(Z; z1 ) = Z 2 − z1 , z1 ∈ C, defines a Weierstrass analytic variety which has exactly one branch, B = {(z1 , z2 ) : z22 = z1 , z1 = 0}. This illustrates both that a branch is connected, but not in general simply connected, and that “above” each point of V there is usually more than one point of B. Note also that A is real-on-real and B ∩ R2 is a real parabola with the origin removed.

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Let E = var (C × C, {h}), h(z1 , z2 ) = A(z2 ; z1 ) and A(Z; z) = Z 2 + z 2 . Note that D({h}) is zero only at 0 ∈ C, that E has two branches, B± = {(z, ±iz) : z ∈ C} and that neither of them is closed under complex conjugation even though E is real-on-real. The next result addresses this issue. L EMMA 7.2.6 Suppose that B is a branch of a real-on-real C-analytic variety var (V × Cn−m , H) and that B ∩ Rn = ∅. Then B ∗ := {(z 1 , · · · , z n ) : (z1 , · · · , zn ) ∈ B} = B. Proof. Since functions in H have Taylor expansions at 0 with   real coefficients and B is a maximal connected set (in V \ var (V, {D(H)}) × Cn−m ) of solutions of the equations h = 0, h ∈ H, the set B ∗ is a maximal in the same sense. By hypothesis, B ∩ B ∗ = ∅. Hence B ∗ = B and the result is proved. The next result shows that the closure of a branch is an analytic variety in a neighbourhood of 0, even though the branch, itself a manifold, need not be. For a branch B,   B denotes B ∩ V × Cn−m , the relative closure of B in V × Cn−m . By Proposition 5.4.2, a Weierstrass analytic variety is the union of the closures (in this sense) of its branches. T HEOREM 7.2.7 Suppose that B is a branch of a Weierstrass analytic variety E = var (V × Cn−m , H) with discriminant D = D(H). Then B = var (V × Cn−m , G) for some finite collection of analytic functions g : V × Cn−m → C. Suppose in addition that H is real-on-real and B ∩ Rn = ∅. Then G is real-onreal, B is a real-on-real C-analytic variety and B ∩ Rn is an R-analytic variety with dimR γ0 (Rn ∩ B) = m. Proof. Let ξ = (z1 , · · · , zm ) ∈ V \ var (V, {D}). Then there are K, say, points of B above ξ. In other words, there are K elements ζj (ξ) ∈ Cn−m , such that (ξ, ζj (ξ)) ∈ B, 1 ≤ j ≤ K. By Remarks 7.2.4 the dependence of ζj (ξ) on ξ is C-analytic locally and therefore, by connectedness, K is independent of ξ ∈ V \ var (V, {D}). Note also that (ξ, ζ) ∈ B if and only if, for all  = (m+1 , · · · , n ) ∈ Cn−m K 

, ζ − ζj (ξ)  = 0.

j=1

  Since this product, as a function of (ξ, ζ) ∈ V \ var (V, {D}) × Cn−m , is independent of permutations of j ∈ {1, 2, · · · , K}, it is a continuous single-valued

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  function of (ξ, ζ) ∈ V \ var (V, {D}) × Cn−m and therefore is C-analytic there. Therefore K 

, ζ − ζj (ξ)  =

j=1



σ g˜σ (ξ, ζ),

(7.2)

σ∈Nn−m 0 |σ|=K

  where the functions g˜σ are analytic on V \ var (V, {D}) × Cn−m . Moreover, for ξ ∈ V \ var (V, {D}), (z1 , · · · , zn ) = (ξ, ζ) ∈ B if and only if g˜σ (z1 , · · · , zn ) = 0 for all σ ∈ Nn−m with |σ| = K. 0 Finally observe that, for all compact sets W ⊂ V , sup{|ζj (ξ)| : ξ ∈ W \ var (V, {D}), 1 ≤ j ≤ K} < ∞. Therefore g˜σ is bounded on (W × Cn−m ) \ var (V × Cn−m , {D}) and, by the Riemann extension theorem 5.4.3, can be extended as an analytic function gσ on all of V × Cn−m . To complete the proof of the first part let G = {gσ : σ ∈ , |σ| = K} and recall that V \ var (V, {D}) is open, dense and connected in Nn−m 0 V. Now suppose that H is real-on-real and B ∩ Rn = ∅. From the implicit function theorem 3.5.4 with F = R, B ∩ Rn is an R-analytic manifold of dimension m. By Lemma 7.2.6, for each j, ζj (ξ) = ζk (ξ) for some k. Therefore the left side of (7.2) is real when , ξ and ζ are real vectors. Therefore g˜σ (ξ, ζ) is real when ξ and ζ are real vectors. This shows that G is real-on-real. Therefore E is a real-on-real C-analytic variety and dimR B ∩ Rn = m. R EMARK 7.2.8 Suppose that m = n − 1 in the preceding theorem. From (7.2) it follows that G has only one element, g say, where g is a polynomial in zn with coefficients analytic on V , its principal coefficient is 1, all the others vanish at 0 ∈ V and its discriminant is not identically zero. Therefore in Theorem 7.2.7 B = var (V × Cn−1 , G) is a Weierstrass analytic variety on V . The following example of a Weierstrass analytic variety in C3 with m = n − 2 shows that this observation may be false when m = n − 1. Let V = C and let E = var (V × C2 , {h, k}) where h(x, y) = y 2 − x3 , k(x, z) = z 2 − x3 , (x, y, z) ∈ C3 . Note that E has two branches B± , B ± = var (V × C2 , {h, k, l± }), where l± = yz ± x3 , (x, y, z) ∈ C3 .

(7.3)

We now show that neither is a Weierstrass analytic variety on V . Suppose that this is false and that B − is a Weierstrass analytic variety defined by yp +

p−1  k=0

Ak (x)y k = 0 and z q +

q−1  l=0

Bl (x)z l = 0,

(7.4)

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where the discriminant of the polynomials is non-zero almost everywhere. Therefore, for almost all x in a neighbourhood of 0 in C, there are exactly pq solutions of (7.4). However, for the same x there are two points (x, y, z) on B − . Hence pq = 2. Suppose p = 1 and q = 2. Then the system y = A0 (x) and z 2 = B1 (x)z + B0 (x) is equivalent to (7.3) with a minus sign. But this is false since (7.3) does not determine y as a function of x. A similar contradiction is reached if q = 1 and p = 2, and for B + . We have seen in Definition 7.2.3, Remark 7.2.4 and Lemma 7.1.6 that γa (B) is irreducible when a ∈ B and B is a branch of a Weierstrass analytic variety. More is true. L EMMA 7.2.9 Let B be a branch of a Weierstrass analytic variety var (V × Cn−m , H). Then γ0 (B) ∈ V0 (Cn ) is irreducible. Proof. The case m = 0 is trivial since B = {0}. Suppose m ∈ N. Suppose that γ0 (B) = α1 ∪ α2 , α1 , α2 ∈ V0 (Cn ). Let E1 , E2 be analytic varieties, defined in a neighbourhood of 0 ∈ Cn , which represents α1 and α2 . Then there exists an open set O in Cn with 0 ∈ O such that B ∩ O = O ∩ (E1 ∪ E2 ). Now for any z ∈ B ∩ O, γz (B) ∈ V(Cn ) is irreducible, by Lemma 7.1.6. Therefore for every point z ∈ B∩O there is an open set Oz in Cn with Oz ∩B ⊂ E1 or Oz ∩ B ⊂ E2 . Since B is a connected analytic manifold it follows from Lemma 7.1.4 that B ⊂ E1 or B ⊂ E2 . Since Ei is closed in V × Cn−m , B ⊂ E1 or B ⊂ E2 . Hence γ0 (B) = γ0 (E1 ) = α1 or γ0 (B) = γ0 (E2 ) = α2 , and γ0 (B) is irreducible. This completes the proof. In Remark 7.2.8, the Weierstrass analytic variety E is defined in terms of Weierstrass polynomials h, k neither of which is the product of two Weierstrass polynomials on V , yet E is not irreducible. The following shows that this does not happen when m = n − 1. L EMMA 7.2.10

Let E denote a Weierstrass analytic variety

var (V × C, {h}), h(z1 , · · · , zn ) = A(zn ; z1 , · · · , zn−1 ). Then γ0 (E) is irreducible if and only if A is not the product of two Weierstrass polynomials on V . Proof. Suppose that γ0 (E) is not irreducible. By Lemma 7.2.9, E has at least two branches and by Remark 7.2.8 the closure of each of these branches is a Weierstrass ˜ has closure defined by a Weierstrass analytic variety. Suppose one such branch B ˜ ˜ polynomial A on V , where A and A are distinct polynomials on V . Then the great˜ for some non-trivial Weierstrass est common divisor of A and A˜ is A˜ and A = AE polynomial E on V (see the last sentence of Remark 7.2.2). Therefore A is the product of two Weierstrass polynomials.

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When A is the product A1 A2 of two Weierstrass polynomials, it is easy to see that γ0 (E) = γ0 (E1 ) ∪ γ0 (E2 ), where Ei are the varieties defined using Aj . Since A is a Weierstrass polynomial, A1 = A2 and therefore γ0 (E) is not irreducible. This completes the proof.

7.3 ANALYTIC GERMS AND SUBSPACES Suppose that α is a C-analytic germ at a with γa (Cn ) = α. Suppose that var (U, G) is a representative of α. Then there is at least one C-analytic function g ∈ G, such that g ≡ 0 on U . Hence there is a complex line segment L through a in U such that g ≡ 0 on L. Since g restricted to L is a complex-analytic function of one complex variable, its zeros are isolated. This shows that there exists a one-dimensional complex linear space Y such that γa (a + Y ) ∩ α = {a}. For the case of real-on-real varieties var (U, G), suppose that 0 ∈ U and that g ∈ G is real-on-real. Then the coefficients in the Taylor expansion of g at 0 are all real and not all zero. Hence there exists a real linear space Yˆ = {tb : t ∈ R}, for some b ∈ Rn , such that γ0 (Yˆ ) ∩ α = {0}. Moreover, g is not identically zero on the complex linear space T = {zb : z ∈ C} and hence γ0 (T ) ∩ α = {0}. We will say that a complex linear subspace T of Cn is a complexified subspace if it has a real basis (see Remark 5.1.2). Equivalently T is complexified if it is closed under complex conjugation of the coordinates of its vectors with respect to the standard real basis. Clearly T = {zb : z ∈ C}, b ∈ Rn , is a complexified space of one complex dimension. Any complexified subspace Z1 of Cn has a (in fact many) complementary complexified subspace Z2 of Cn such that Cn = Z1 ⊕ Z2 . This ensures that the choice of basis in the last part of the next lemma is possible. L EMMA 7.3.1 Suppose that α ∈ V0 (Cn ), n ≥ 2, and Y is a linear subspace of Cn such that α ∩ γ0 (Y ) = {0}. Choose a basis of Cn such that Y = {(0, · · · , 0, zm+1 , · · · , zn ) : (zm+1 , · · · , zn ) ∈ Cn−m }

(7.5)

and let P denote the projection onto Cm given by first m coordinates, so that P (E) = {(z1 , · · · , zm ) : (z1 , · · · zn ) ∈ E},

(7.6)

where E is a representative of α. Then γ0 (P (E)) ∈ V(Cm ). If Y is a complexified subspace, α is real-on-real and we choose a real basis of Cn such that (7.5) holds. Then γ0 (P (E)) is real-on-real. Proof. The cases m = 0 (α = γ0 (Cn ), Y = {0}) and m = n (α = {0}, Y = Cn ) are trivial  and we supposethroughout that 0 < m < n. In the coordinates (7.5), let α = γ0 var (W × Bδ (C) , {g1 , · · · , gν } , ν ∈ N, for a small positive δ where W = {(z1 , · · · , zn−1 ) ∈ Cn−1 : |zj | < δ, 1 ≤ j ≤ n − 1}. Since {(0, · · · , 0, z) ∈ Cn : z ∈ C} ⊂ Y and γ0 (Y ) ∩ α = {0},

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gk (0, · · · , 0, zn ) ≡ 0 for some k when zn ∈ Bδ (C). Relabelling {g1 , · · · , gν } if necessary, suppose that k = 1. By the Weierstrass preparation theorem 5.3.1, there is no loss of generality in supposing that g1 is given by a polynomial A1 of the form (5.1). If A1 has degree p, say, then, by the Weierstrass division theorem 5.2.1, we may suppose without loss of generality that each of the other gk , k ≥ 2, in the definition of α, has the form gk (z1 , · · · , zn ) = Ak (zn ; z1 , · · · , zn−1 ) where Ak (Z; z1 , · · · , zn−1 ) isa polynomial on W of degree at most p − 1. Thus var (W ×Bδ (C), {g1 , · · · , gν } is a representative of α and the family {g1 , · · · , gν } of analytic functions satisfies the hypotheses of the projection lemma, Theorem 6.2.5. Therefore the projection of var (W × Bδ (C), {g1 , · · · , gν }) onto Cn−1 × {0} is an analytic variety in Cn−1 . Let β ∈ γ0 (Cn−1 ) denote its germ and let Yˆ = {(0, · · · , 0, zm+1 , · · · , zn−1 ) : (zm+1 , · · · , zn−1 ) ∈ Cn−m−1 }. Since α ∩ γ0 (Y ) = {0} in Cn , γ0 (Yˆ ) ∩ β = {0} in Cn−1 . We can now repeat the argument n − m times to prove the first part of the lemma. In the case when Y is a complexified subspace, choose a real basis for Cn such that (7.5) holds. Then the projection lemma at each step gives a real-on-real variety. This completes the proof. L EMMA 7.3.2 Let Y be a linear subspace which is maximal with respect to α ∈ V0 (Cn ) in the sense that for any linear subspace Y˜ of Cn γ0 (Y ) ∩ α = {0} and Y˜ = Y ⊂ Y˜ implies that γ0 (Y˜ ) ∩ α = {0}.

(7.7)

Let n ≥ 2, m = n − dim Y ∈ {1, · · · , n − 1}, and choose coordinates (7.5). Let E be any representative of α. Then γ0 (P (E)) = γ0 (Cm ), where P (E) is defined in (7.6). Proof. We have seen in the previous lemma that P (E) is an analytic variety in Cm . Suppose that at 0 its germ β = γ0 (Cm ). Then there exists a non-trivial linear space L ⊂ Cm such that β ∩ γ0 (L) = {0}. Let Y˜ = L × Y . Then γ0 (Y˜ ) ∩ α = {0} which violates the maximality of Y . This proves the lemma. This is false for real analytic germs as the real germ of {(x, y) ∈ R2 : y − x2 = 0} illustrates. L EMMA 7.3.3 Suppose that α ∈ V0 (Cn ) is real-on-real, n ≥ 2, and T is a complexified subspace of Cn such that α ∩ γ0 (T ) = {0}. Suppose that T is a maximal complexified subspace in the sense that if T˜ is a complexified space then γ0 (T ) ∩ α = {0} and T ⊂ T˜ = T implies that γ0 (T˜) ∩ α = {0}.

(7.8)

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With respect to a real basis such that T = {(0, · · · , 0, zm+1 , · · · , zn ) : (zm+1 , · · · , zn ) ∈ Cn−m },

(7.9)

m

γ0 (P (E)) = γ0 (C ). Proof. We have seen in Lemma 7.3.1 that in this case P (E) is a real-on-real analytic variety in Cm . Suppose that its germ at 0, β = γ0 (Cm ). Then, by the remarks at the beginning of the section, there exists a non-trivial complexified subspace L ⊂ Cm such that β ∩ γ0 (L) = {0}. Let T˜ = L × T . Then γ0 (T˜) ∩ α = {0} which violates the maximality of T . This proves the lemma. E XAMPLE 7.3.4 The following is an illustration of (7.7) and (7.8). Let g, h : C4 → C be defined by g(w, x, y, z) = y 2 + z 2 , h(w, x, y, z) = x, (w, x, y, z) ∈ C4 . Then E is real-on-real and E ∩ R4 = spanR {(1, 0, 0, 0)} where E = var (C4 , {g, h}) = {(w, 0, y, z) ∈ C4 : y 2 + z 2 = 0},   and α ∩ γ0 {0} × R3 = {0}, where α = γ0 (E). However the three-dimensional complexified space T = {0} × C3 is not maximal in the sense of (7.8); it is too big. It is easy to see that each of the two-dimensional complexified spaces {0} × C × {0} × C and {0} × C × C × {0} are maximal in the sense of (7.7) and (7.8). 7.4 GERMS OF C-ANALYTIC VARIETIES We are now in a position to show that if α is a C-analytic germ at 0 there exists a Weierstrass analytic variety E, a subset C and a branch B of E such that α = γ0 (C) and γ0 (B) ⊂ α.   Suppose that α ∈ V0 (Cn ). If n = 1 then α ∈ ∅, {0}, γ0 (C) . If {0} ⊂ α but  α∈ / {0}, γ0 (Cn ) then n ≥ 2. Moreover, since α = γ0 (Cn ), §7.3 ensures the existence of a non-trivial linear subspace Y of Cn such that γ0 (Y ) ∩ α = {0}

(7.10)

and, since α = {0}, we infer that Y = Cn . Let Y be such a linear subspace, n ≥ 2, m = n − dim Y ∈ {1, · · · , n − 1}

(7.11)

and choose coordinates such that Y = {(0, · · · , 0, zm+1 , · · · , zn ) : (zm+1 , · · · , zn ) ∈ Cn−m }. (7.12)   T HEOREM 7.4.1 Let α ∈ V0 (Cn ) \ {0}, γ0 (Cn ) , n ≥ 2, and choose coordinates such that (7.10), (7.11) and (7.12) hold. Then there exists a Weierstrass  analytic variety var V × Cn−m , H , 1 ≤ m < n, such that   α ⊂ γ0 var (V × Cn−m , H) .

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Proof. As in the proof of Lemma 7.3.1, the Weierstrass preparation 5.3.1 and division 5.2.1 theorems,and Proposition 6.2.4 can be used to reduce the problem to the case when α = γ0 var (W × C, G) , where G = {g1 , · · · , gν }, ν ∈ N, and g1 (z1 , · · · , zn ) = znp + ap−1 (z1 , · · · , zn−1 )znp−1 + · · · + a0 (z1 , · · · , zn−1 ), aj (0) = 0, 0 ≤ j ≤ p − 1, D(g1 ) ≡ 0 on W, and each gk , k ≥ 2, is a polynomial in the same variable zn , with coefficients that are C-analytic functions of the other variables; g1 is the only polynomial in G with highest degree p. First let ν = 1 and G = {g1 } where 2 ≤ n ∈ N is arbitrary. Since g1 is a Weierstrass polynomial in zn , Proposition 6.1.1 gives that  α = γ0 {(z1 , · · · , zn−1 ) ∈ Cn−1

 : (z1 , · · · , zn ) ∈ var (W × C, {g1 })} = γ0 (Cn−1 ).

By (7.10) and (7.12), m = n − 1 and the theorem holds with H = {g1 } and  V = W . (In fact we get α = γ0 var (V × C, H) in this case.) For the general case when G = {g1 , · · · , gν } we argue by induction on n ≥ 2. The inductive hypothesis is that for all n ≥ 3 and all α ˆ ∈ V0 (Cnˆ ), 2 ≤ n ˆ < n, the conclusion of the theorem holds with m, n, Y replaced with m, ˆ n ˆ , Yˆ satisfying ˆ = {0}, n ˆ ≥ 2, m ˆ =n ˆ − dim Yˆ ∈ {1, · · · , n ˆ − 1} γ0 (Yˆ ) ∩ α ˆ Yˆ = {(0, · · · , 0, zm+1 , · · · , znˆ ) : (zm+1 , · · · , znˆ ) ∈ Cnˆ −m }. ˆ ˆ This hypothesis has been verified when n= 3 because then n ˆ = 2, m ˆ = 1. We no longer need to consider the case α = γ0 var (W × C, {g1 } . So suppose n ≥ 3 and, for some ν ≥ 2,   α = γ0 var (W × C, {g1 , · · · , gν }) , where the set G = {g1 , · · · , gν } satisfies the hypotheses of the projection lemma, Theorem 6.2.5. Let α ˆ = γ0 (A), where A ⊂ W ⊂ Cn−1 denotes the set given by the projection lemma, and note that α ˆ ∈ V0 (Cn−1 ) by Remark 7.1.2. Suppose that α ˆ = {0}. Then in a sufficiently small neighbourhood of the origin in Cn , (z1 , · · · , zn ) ∈ var (W × C, G) only if znp = g1 (0, 0, · · · , zn ) = 0. It now follows that α = {0} which contradicts the hypothesis of the theorem. Hence α ˆ = {0}. Now suppose that α ˆ = γ0 (Cn−1 ). It follows from (7.10) and (7.12) that n − 1 = m. Let H = {g1 }, V = W. Since α ⊂ γ0 (var (V × C, H)), the theorem holds in this case.

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  Finally we come to the case α ˆ∈ / {0}, γ0 (Cn−1 ) , n ≥ 3, m < n − 1. Let Yˆ = {(z1 , · · · , zn−1 ) : z1 = · · · = zm = 0}, where m is defined in (7.10) and (7.12). It follows from the definition of A and (7.12) that γ0 (Yˆ ) ∩ α ˆ = {0}. With n ˆ = n − 1, m ˆ = m, the inductive hypothesis gives that the theorem holds in Cn−1 , n ≥ 3. Thus, in the same coordinates, there exists (a possibly smaller) δ > 0, a set V = {(z1 , · · · , zm ) ∈ Cm : |z1 |, · · · , |zm | < δ}, ˆ = {Aˆm+1 , · · · , Aˆn−1 } of Weierstrass polynomials on V with and a collection H ˆ discriminant D = D(H) not identically zero and ˆ γ0 (A) ⊂ γ0 (V × Cn−m−1 , H). Let

 Υ(z1 , · · · , zm ) = (ˆ zm+1 , · · · , zˆn−1 ) ⊂ Cn−m−1 :  Aˆj (ˆ zj ; z1 , · · · , zm ) = 0, m + 1 ≤ j ≤ n − 1 .

ˆ ≡ 0, the dependence of the zˆm+j on (z1 , · · · , zm ) ∈ V \ var (V, D), ˆ an Since D open, dense, connected subset of V , is locally C-analytic, by the analytic implicit ˆ by function theorem 4.5.4. Now define a polynomial on V \ var (V, D) Aˆn (Z; z1 , · · · , zm ) =



A1 (Z; z1 , · · · , zm , zˆm+1 , · · · , zˆn−1 ).

(ˆ zm+1 ,··· ,ˆ zn−1 ) ∈Υ(z1 ,··· ,zm )

By choosing a smaller value of δ in the definition of V if necessary we see that the coefficients of Aˆn are bounded and hence, by the Riemann extension theorem 5.4.3, can be extended as C-analytic functions to all of V . Note that in Aˆn the coefficient of the highest power of Z is 1 and that all the other coefficients vanish at 0 ∈ V . After simplification (Remark 7.2.2) Aˆn becomes a Weierstrass polynomial on V . ˆ ∪ {Aˆn }, a collection of n − m Weierstrass polynomials on V . Let Let H = H D(z1 , · · · , zm ) denote the product of their discriminants, which is non-zero on an open dense connected subset of V . Now var (V ×Cn−m , H) is a Weierstrass analytic variety. Suppose (z1 , · · · , zn ) belongs to a representative of α in a sufficiently small neighbourhood of 0. Then (z1 , · · · , zn−1 ) ∈ A, (z1 , · · · , zm ) ∈ V, (zm+1 , · · · , zn−1 ) ∈ Υ(z1 , · · · , zm ), g1 (z1 , · · · , zn ) = 0. Thus

This completes the proof.

  α ⊂ γ0 var (V × Cn−m , H) .

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T HEOREM 7.4.2 Let α = γ0 (E) in the preceding theorem and with P as in Lemma 7.3.1 suppose γ0 (P (E)) = γ0 (Cm ). Then γ0 (B) ⊂ α for some branch B of the Weierstrass analytic variety var (V × Cn−m , H) in Theorem 7.4.1. Proof. Without loss of generality suppose that E = var (V × Cm−n , G), where G is a finite collection of analytic functions and V × Cn−m is as in Theorem 7.4.1. With H as in the conclusion of this theorem, let O ⊂ V \ var (V, (D(H))) be a non-empty open ball on which the discriminant D(H) is nowhere zero. For any ξ ∈ O, we may write 

 {ξ} × Cn−m ∩ var (V × Cn−m , H) = {(ξ, ζj (ξ)) : 1 ≤ j ≤ p},

where p is the product of the degrees of the Weierstrass analytic polynomials in H, and the ζj are analytic functions from O into Cn−m , as in Remark 7.2.4. According to Lemma 7.1.3, every open set Oj = {ξ ∈ O : (ξ, ζj (ξ)) ∈ / E} is either empty or dense in O. However, by hypothesis, ∩pj=1 Oj is empty. Therefore at least one of these sets, Oj0 , is empty. In other words the analytic manifold Mj0 = {(ξ, ζj0 (ξ)) : ξ ∈ Oj0 }   is a subset of E. By Lemma 7.1.4, B ∩ (V × Cn−m ) ⊂ E, where B is the branch which contains Mj0 and the proof is complete. C OROLLARY 7.4.3 From the maximality hypotheses of Lemmas 7.3.2 and 7.3.3, the conclusion on Theorem 7.4.2 holds. Proof. This follows by combining the lemmas and the theorem cited. C OROLLARY 7.4.4 (Dimension of α) (a) Suppose n ≥ 2, the hypotheses of Lemma 7.3.2 hold, and consequently, in the notation of Theorem 7.4.2,   γ0 (B) ⊂ α ⊂ γ0 var (V × Cn−m , H) .

(7.13)

  Also, var V × Cn−m , H contains no manifold of dimension larger than m = dimC α, and n − m = max{dim Y : γ0 (Y ) ∩ α = {0}, Y ⊂ Cn a linear space}.

(7.14)

The right hand side of (7.14) is called the codimension of the analytic germ α. Moreover var (V × Cn−m , H ∪ {D(H)}) contains no manifold of dimension equal to or greater that m. (b) If the hypotheses of Lemma 7.3.3 hold and m is defined there instead, then m = dimC α and the conclusion of part (a) is valid. In this case the dimension of α is equal to m whether defined in terms of maximal complex subspaces as in part (a), or of maximal complexified spaces as in part (b).

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Proof. It is clear from (7.13) that dimC B = m ≤ dimC α. If D(H) is nowhere zero, or zero only at 0 ∈ V , it is easy to see that E = var (V × Cn−m , H) contains no manifold of dimension larger  than m and hence dimC α = m. Suppose that {0} ⊂ γ0 (var (V, {D(H)})) ∈ / {0}, γ0 (Cm ) and let N be a manifold of dimension strictly greater than m which is a subset of E. If x ∈ B ∩ N for some branch B then a neighbourhood of x in N is a subset of a neighbourhood of x in B, which is impossible since B is an m-dimensional manifold. Therefore B ∩ N = ∅ for all branches B of E. In other words N ⊂ var (V × Cn−m , H ∪ D(H)). Let Pj (ξ) denote the j th coordinate of ξ ∈ Cn or Cm . Then Pj is an analytic function and §7.3 gives the existence of coordinates on V such that   γ0 V, {D(H), P1 , · · · , Pm−1 } = {0}, so that

  γ0 var V × Cn−m , H ∪ {D(H)} ∩ γ0 (Y ) = {0}

where Y = {(z1 , · · · , zn ) ∈ Cn : z1 = · · · = zm−1 = 0}. Therefore Theorem 7.4.1 yields the existence of a Weierstrass analytic variety ˜ ˜ , H)), m ˜ < m, such that var (V˜ × Cn−m ˜ ˜ , H). N ⊂ var (V × Cn−m , H ∪ {D(H)}) ⊂ var (V˜ × Cn−m

Repeated finitely often we find that this holds with m ˜ = 1, which is impossible. Hence var (V × Cn−m , H) contains no analytic manifold of dimension larger than m, dimC α = m and var (V × Cn−m , H ∪ {D(H)}) contains no manifold of dimension m or more. That (7.14) holds follows from Corollary 7.4.3. The proof of (b) is the same once Lemma 7.3.3 is taken into account. Now we improve slightly on the observation in Lemma 7.2.9 that γ0 (B) is irreducible when B is a branch of a Weierstrass analytic variety. L EMMA 7.4.5 Let B be a branch of a Weierstrass analytic variety var (V × Cn−m , H) and suppose that α ∈ V0 (Cn ) is such that γ0 (B) = α ⊂ γ0 (B). Then dimC α < m. Proof. Suppose that γ0 (B) = α ⊂ γ0 (B), and let E = var (U, G) ⊂ B ∩ (V × Cn−m ), where U is an open set with B ∪ {0} ⊂ U and such that α = γ0 (E). Let D(H) denote the discriminant of H on V and suppose that dimC α ≥ m. We will infer that γ0 (B) = α, a contradiction which will prove the lemma. Define an analytic manifold M ⊂ E as consisting of all (dimC α)-regular points of E. If M ⊂ var (V × Cn−m , H ∪ {D(H)}), then dimC α < dimC B = m, by Lemma 7.4.4. Since this is false, by assumption, M ∩ B = ∅ and dimC α = dimC M = m. Therefore there exists a point z ∈ M ∩ B which has a neighbourhood Oz in B which is a subset of M . From Lemma 7.1.4 and the fact that B is connected it follows that B ⊂ E. This contradiction proves the result.

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C OROLLARY 7.4.6

Suppose that m = 1 in Theorem 7.4.1. Then

 α = γ0 {0} ∪γ0 (B)⊂α B ,

where the union is over all branches B of var (V × Cn−1 , H) with germ contained in α. Proof. Since m = 1, V ⊂ C and there is no loss of generality in assuming that the discriminant D(H) is non-zero on V \{0} and that B = B∪{0} for all branches B. Now suppose that B is a branch of var (V × Cn−1 , H) such that α ∩ γ0 (B) = {0} and that γ0 (B) ⊂ α. Then γ0 (B) ⊂ α and α ∩ γ0 (B) is an analytic germ in Cn with γ0 (B) = α ∩ γ0 (B) ⊂ γ0 (B).   Therefore, by Lemma 7.4.5, dimC α ∩ γ0 (B) = 0. Hence α ∩ γ0 (B) = {0}. Since this is false we conclude that γ0 (B) ⊂ α for every branch B of var (V × C, H) with γ0 (B) ∩ α ⊂ {0}. This proves the corollary. We are now in a position to say something, but not everything, about the structure of C-analytic varieties. The following extension of the preceding corollary is more than sufficient for our purposes. T HEOREM 7.4.7 (A Structure Theorem) Let n ≥ 2 and α ∈ V0 (Cn ) \ {0} be such that {0} ⊂ α = γ0 (Cn ). Then there exist sets B1 , · · · , BN , such that   (a) α = γ0 B1 ∪ · · · ∪ BN ∪ {0} . (b) Each Bj , 1 ≤ j ≤ N , after a linear change of coordinates (depending on j), is a branch of a Weierstrass analytic variety (depending, including its dimension, on j). (c) dimC α = max1≤j≤N {dimC Bj }. (d) If L ⊂ Cn , γ0 (L) = ∅, is a connected C-analytic manifold of dimension l ∈ {1, · · · , n} the points of which are l-regular points of a representative of α, then there exists j ∈ {1, · · · , N } such that γ0 (L) ⊂ γ0 (Bj ) and dimC Bj = l. (e) If α is real-on-real, then it can be arranged that each branch Bj with Bj ∩ Rn = ∅ is real-on-real.   ˜1 ∪· · ·∪ B ˜K ∪{0} where the B ˜j denotes those branches (f) α∩γ0 (Rn ) = γ0 B which intersect Rn non-trivially. ˜j ∩ Rn ). (g) dimR (α ∩ Rn ) = max1≤j≤K dimR (B

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Proof. We use induction on the dimension of α. Suppose that var (V × Cn−m , H) and the coordinate system are given by Theorem 7.4.2. Consider first all the branches of var (V × Cn−m , H) such that γ0 (B) ⊂ α. If m = 1, Corollary 7.4.6 shows that these are all the branches that we need for the result to hold. Suppose m ≥ 2 and make the inductive hypothesis that the results (a)-(d) of the theorem hold for all smaller values of m. (a)-(c) According to Theorem 7.4.1,   α ⊂ γ0 var (V × Cn−m , H) = ∪B, where the union is over all branches B of var (V × Cn−m , H). In addition to the ˜ of var (V × Cn−m , H) such that branches B with γ0 (B) ⊂ α, consider branches B ˜ ∩ α = γ0 (B). ˜ ∅=  γ0 (B) ˜ has dimension strictly smaller than Since, by Lemma 7.4.5, the germ α ∩ γ0 (B) m, we can apply the inductive hypothesis to each of these branches to complete the proof of (a)-(c). n−m m = dimC α and that . (d) Note that  we may suppose L ⊂ V ×C  dimC L ≤n−m If γ0 (L) ⊂ γ0 var (V × C , H ∪ {D(H)}) , we apply the inductive hypothesis to obtain the required result.   If γ0 (L) ⊂ γ0 var (V × Cn−m , H ∪ {D(H)}) , then L ∩ B = ∅ for at least one branch B of var (V × Cn−m , H). For all z ∈ L ∩ B, L and B coincide locally, in a neighbourhood of z and dimC L = dimC B. By Theorem 7.2.7 B ∩ (V × Cn−m ) is an analytic variety. That γ0 (L) ⊂ γ0 (B) now follows from Lemma 7.1.4. (e) It is clear that if α is real-on-real, and maximal complexified spaces are used, as in Lemma 7.3.3, to choose coordinates, then the branches Bj which emerge are real-on-real. Parts (f) and (g) follow from the second part of Theorem 7.2.7.   C OROLLARY 7.4.8 Let α ∈ V0 (Cn ) \ ∅, {0}, γ0 (Cm ) be irreducible. Then (possibly after a linear change of coordinates) α = γ0 (B) where B is a branch of some Weierstrass analytic variety. If α is real-on-real and α ∩ γ0 (Rn ) = {0}, then B is a branch of a real-on-real variety. Proof. In the notation of Theorem 7.4.7, α = γ0 (B1 ) ∪ · · · ∪ γ0 (BN ), and since α is irreducible the result follows. The following example shows that when α is an irreducible C-analytic variety it does not necessarily follow that α ∩ γ0 (Rn ) is an irreducible real analytic variety. E XAMPLE 7.4.9

Let V = C2 and E = var (V × C, {h}) where

h(x, y, z) = z 2 + x2 y 2 (x2 + y 2 ), (x, y, z) ∈ V × C. Clearly E ⊂ C3 is a Weierstrass analytic variety defined by the polynomial A(Z; x, y) = Z 2 + x2 y 2 (x2 + y 2 ), (x, y, z) ∈ V × C.

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If A is a product of two Weierstrass polynomials then each has order one, and it is easily checked that this is impossible. Hence, by Lemma 7.2.10, γ0 (E) is an irreducible C-analytic variety. However E ∩ R3 coincides with {(x, y, z) ∈ R3 : x = 0, z = 0} ∪ {(x, y, z) ∈ R3 : y = 0, z = 0}. Therefore γ0 (E ∩ Rn ) is not a real irreducible variety.

7.5 ONE-DIMENSIONAL BRANCHES The following results are aspects of the theory of Puiseux series sufficient for our later needs. T HEOREM 7.5.1 Suppose that m = 1, 2 ≤ n ∈ N and that B is a branch of the Weierstrass analytic variety E = var (V × Cn−1 , H) where V is chosen so that D(H) is non-zero on V \ {0}. Then there exist K ∈ N, δ > 0 and a C-analytic function ψ : {z ∈ C : |z|K < δ} → Cn−1 such that the mapping z → (z K , ψ(z)) is injective, ψ(0) = 0 and {0} ∪ B = B ∩ (V × Cn−1 ) = {(z K , ψ(z)) : |z|K < δ}. R EMARK 7.5.2 A function ψ satisfying the conclusion of the theorem is not unique. Indeed, if ψ satisfies the theorem and ζ = 1 is a K th root of unity then ˜ ψ(z) := ψ(ζz) defines another function which also satisfies the conclusion of the theorem. Proof. Let H = {h2 , · · · , hn } where hk (z1 , · · · , zn ) = Ak (zk ; z1 ), and each Ak is a Weierstrass polynomial of degree pk , say, 2 ≤ k ≤ n. If the discriminant D(H) is not zero at z1 = 0, then, for all k ∈ {2, · · · , n}, Ak (Z; z1 ) = Z − ak (z1 ) where ak , is an analytic function on V ⊂ C. In this case the theorem holds with K = 1 and ψ(z1 ) = (a2 (z1 ), a3 (z1 ), · · · , an (z1 )), z1 ∈ V. Now suppose that D(H) is zero at 0. Note that for z1 ∈ V \ {0} each of the polynomials Ak (Z; z1 ) has only simple roots. Let Vˆ denote the half-plane in C defined by Vˆ = {z ∈ C : z = ρ + iθ, − ∞ < ρ < log δ, θ ∈ R}, where δ is given in the definition of V , and let ˆ k (z, zk ) = Ak (zk ; ez ), z ∈ Vˆ , zk ∈ C. h

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Let ˆ 2, · · · , h ˆ n } and E ˆ = {h ˆ = var (Vˆ × Cn−1 , H). ˆ H ˆ is a branch of E, ˆ where It is clear that B is a branch of E if and only if B ˆ B = {(ez , ξ) : (z, ξ) ∈ B}, ξ ∈ Cn−1 . ˆ is nowhere zero on Vˆ and every Since D(H) is nowhere zero on V \ {0}, D(H) ˆ point of E is 1-regular (Definition 7.1.1). We can therefore write   ˆ = {(z, ξq (z)) : 1 ≤ q ≤ p}, {z} × Cn−1 ∩ E (n where p = k=2 pk . By the analytic implicit function Theorem 4.5.4, each ξq is defined locally on Vˆ as a C-analytic function with values in Cn−1 and, since Vˆ is ˆ is the union of the simply connected, they define analytic functions on Vˆ . Thus E disjoint graphs of the functions ξq : Vˆ → Cn−1 , 1 ≤ q ≤ p. Recall that, for z ∈ Vˆ , each component of ξq (z) ∈ Cn−1 is a simple root of a polynomial Ak (Z; ez ), 2 ≤ k ≤ n. Therefore the set-valued map   z → (ez , ξq (z)) : 1 ≤ q ≤ p is 2πi-periodic on Vˆ . Moreover if, for some zˆ ∈ Vˆ and some m ∈ Z, ξq1 (ˆ z ) = ξq2 (ˆ z + 2πmi), q1 , q2 ∈ {1, · · · p}, then ξq1 (z) = ξq2 (z + 2πmi) for all z ∈ Vˆ , by the analytic implicit function theorem 4.5.4 and analytic continuation. Therefore, for each q ∈ {1, · · · , p}, the mapping z → (ez , ξq (z)) ∈ E, z ∈ Vˆ ,

(7.15)

is periodic with period 2πKq i and is injective on the set Vq = {z = ρ + iθ ∈ Vˆ : 0 < θ ≤ 2πKq }, for some Kq ∈ {1, · · · , p}. It is easy to see that its image on Vq is both open and closed in E and hence is a branch of E. For a given branch B, choose q such that the image of (7.15) on Vq coincides with B. We have seen that an injective parameterization of B is given by   B = (ez , ξq (z)) : z ∈ Vq . Since z → ξq (Kq z) has period (not necessarily minimal) 2πi, we can define an analytic function ψ˜ : {z : 0 < |z| < δ 1/Kq } → C by ˜ 1 ) = ξq (Kq log z1 ) ψ(z where it does not matter which branch of log is chosen. Thus ˜ z ), Kq z ∈ Vˆ . ξq (Kq z) = ψ(e

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This gives a new injective parameterization of B, namely  K  ˜ 1 )) : 0 < |z| < δ 1/Kq , B = (z1 q , ψ(z ˜ 1 ) = 0. The Riemann extension theorem 5.4.3 where ψ is analytic and limz1 →0 ψ(z ˜ means that ψ has an analytic extension ψ defined on the ball {z1 ∈ C : |z1 | < δ 1/Kq } with ψ(0) = 0. Let K = Kq to complete the proof. C OROLLARY 7.5.3 In Theorem 7.5.1 suppose γ0 (B ∩ Rn ) ∈ / {∅, {0}}. Then there exists k ∈ N0 with 0 ≤ k ≤ 2K − 1 such that    (7.16) Rn ∩ B = (−1)k rK , ψ(r exp(kπi/K)) : −δ 1/K < r < δ 1/K , and this parameterization is injective. / {{0}, ∅} there exists, by Theorem 7.5.1, a sequence Proof. Since γ0 (B ∩ Rn ) ∈ {zj } ⊂ C with zj → 0 such that zjK ∈ R and ψ(zj ) ∈ Rn−1 for all j ∈ N. Therefore, without loss of generality we may assume, for some k ∈ {0, 1, · · · , 2K − 1}, that zj = |zj | exp(kπi/K) and ψ(|zj | exp(kπi/K)) ∈ Rn−1 for all j ∈ N. Since ψ is a C-analytic function of one complex variable we can infer that ψ(r exp(kπi/K)) ∈ Rn−1 for all r ∈ R with − δ 1/K < r < δ 1/K . If there exists l ∈ {0, 1, · · · , 2K − 1} different from k and a sequence ρj > 0 such that ψ(ρj exp(lπi/K)) is real and ρj → 0 as j → ∞, then, by the preceding argument, we may assume that ψ(r exp(lπi/K)) is real for all r with −δ 1/K < r < δ 1/K . We will now show that l − k ∈ KZ. Suppose that this is false. For p ∈ N,  dp ψ  r exp(kπi/K)  drp r=0 = exp(pπi(k − l)/K)

 dp ψ  r exp(lπi/K)  , drp r=0

and the derivatives are real. Therefore, for all p with p(l − k) ∈ / KZ, it follows that dp ψ (0) = 0. dz p Let p0 ∈ N be the generator of the ideal {p ∈ Z : p(l − k) ∈ KZ}. Then the power series expansion of ψ(z) at z = 0 involves only powers of z p0 and it follows that ψ(z1 ) = ψ(z2 ) for all z1 , z2 ∈ C such that z1p0 = z2p0 . Since p0 divides K, z1K = z2K for all such z1 , z2 . Therefore if z1p0 = z2p0 ,  K    z1 , ψ(z1 ) = z2K , ψ(z2 ) .

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Now the injectivity in the theorem above gives that p0 = 1 and k − l ∈ KZ. This completes the proof. L EMMA 7.5.4 Suppose the Weierstrass polynomials which define the Weierstrass analytic variety var (V × Cn−1 , H) in Theorem 7.5.1 are real-on-real and that the discriminant D(H) is non-zero on V \ {0} ⊂ C. Then B ∩ Rn ∈ / {∅, {0}} implies that γ0 (B ∩ Rn−1 ) ∈ / {∅, {0}}. ˆ2 , · · · , x ˆn ) ∈ (Rn ∩ B) \ {0}. Then each x ˆk , k ≥ 2, is Proof. Suppose that (ˆ x1 , x ˆ1 ∈ R, of a polynomial whose coefficients (C-analytic a simple root, when z1 = x functions of z1 ) are real when z1 = x1 ∈ R, and are zero when z1 = 0. From the real implicit function theorem 3.5.4 it follows that each of the polynomials in H has a real root xk ∈ R, 2 ≤ k ≤ m, which is a real-valued analytic function of x1 when x ˆ1 x1 > 0 and x1 is sufficiently small. Moreover, xk → 0 as x1 → 0 by Lemma 6.1.1. It follows from Theorem 7.5.1 that there exists k ∈ {0, 1, · · · , 2K − 1} such that ψ(r exp(kπi/K)) ∈ Rn−1 for − δ 1/K < r < δ 1/K .

E XAMPLE 7.5.5 nomials

Consider the collection H of three Weierstrass analytic polyZ 2 − z1 , Z 3 − z12 , Z 4 − z13 ,

and let V be the disc of radius 2 with centre 0 in C. Then the corresponding Weierstrass analytic variety, var (V × C3 , H)   = (z1 , z2 , z3 , z4 ) : |z1 | < 2, z22 − z1 = z33 − z12 = z44 − z13 = 0 , has two branches. To see this note that the branch which contains (1, 1, 1, 1) also contains the closed Jordan curve Γ1 = {(eit , e(it/2) , e(2it/3) , e(3it/4) ) : t ∈ [0, 24π] }. Γ1 projects onto the unit circle in V and contains 12 points above 1 ∈ C. Similarly the branch B2 containing (1, −1, 1, 1) contains the closed Jordan curve Γ2 = {(eit , ei(π+t/2) , e(2it/3) , e(3it/4) ) : t ∈ [0, 24π] }. Γ2 also has 12 points above 1 ∈ C and projects onto the unit circle in V . Since the equations t = s mod 2π, 2t/3 = 2s/3 mod 2π,

π + t/2 = s/2 mod 2π, 3t/4 = 3s/4 mod 2π,

ANALYTIC VARIETIES

99

imply that 2k = 4l − 2 = 3m = 8n/3 for some k, l, m, n ∈ Z, they have no solutions. Therefore Γ1 ∩Γ2 = ∅. Since there are at most 2×3×4 = 24 points of var (V × C3 , H) above 1 ∈ V there are at most two branches of this variety. Thus the variety has exactly two branches and K = 12 in Theorem 7.5.1. Moreover candidates for the function ψ (remember it is not unique) corresponding to these branches, are ψ1 (z) = (z 6 , z 8 , z 9 ) and ψ2 = (z 6 , z 8 , −iz 9 ). Moreover B1 ∩ Rn = {(r12 , r6 , r8 , r9 ) : r ∈ (−21/12 , 21/12 )} B2 ∩ Rn = {(r12 , −r6 , r8 , r9 ) : r ∈ (−21/12 , 21/12 )}, which correspond to k = 0 and k = 6, respectively, in Corollary 7.5.3. Note that Lemma 7.5.4 is false if the coefficients of elements of H are complex when their argument is real. For example, when n = 2 let h(Z, z1 ) = Z 2 − (1 + i)z1 − iz12 . Then var (V, {h}) ∩ R2 = {(0, 0), (1, 1), (1, −1)}.

7.6 NOTES ON SOURCES This material is to be found in the books Chirka [18], Federer [29], Mumford [45] and Narasimhan [46]. For the theory of analytic varieties far beyond our needs, see Lojasiewicz [43].

Chapter Eight Local Bifurcation Theory In this chapter we consider the existence of solutions to nonlinear equations of the form F (λ, x) = 0 where F : F × X → Y , F (λ, 0) = 0 ∈ Y for all λ ∈ F and X and Y are Banach spaces. A solution is a pair (λ, x) and the parameter is not prescribed a priori. For the moment we treat the cases of differentiable F and analytic F simultaneously. Local bifurcation theory addresses the question: for which λ0 ∈ F is there a sequence {(λn , xn )} ⊂ F × (X \ {0}) of solutions converging to (λ0 , 0) in F × X? Such (λ0 , 0) are called bifurcation points on the line of trivial solutions {(λ, 0) : λ ∈ F}. For obvious reasons, λ0 ∈ F is sometimes referred to as the bifurcation point.

8.1 A NECESSARY CONDITION Suppose that F is continuously differentiable from a neighbourhood of (λ0 , 0) in F × X into Y . If ∂x F [(λ0 , 0)] is a homeomorphism, the implicit function Theorem 3.5.4 says that in a neighbourhood of (λ0 , 0) in F×X all the solutions to F (λ, x) = 0 lie on a unique curve {(λ, x) : x = φ(λ), λ ∈ (λ0 − , λ0 + )}, for some > 0. Since the line of trivial solutions passes through (λ0 , 0), we conclude that φ(λ) = 0 for all λ ∈ (λ0 − , λ0 + ). Hence, when F is continuously differentiable at (λ0 , 0), a necessary condition for (λ0 , 0) to be a bifurcation point is that ∂x F [(λ0 , 0)] : X → Y should not be a homeomorphism. Note that, because X and Y are Banach spaces and since, by definition, ∂x F [(λ0 , 0)] : X → Y is bounded, this is equivalent to the weaker statement that ∂x F [(λ0 , 0)] : X → Y should not be a bijection (see Corollary 2.4.3). This latter condition is often much easier to verify in particular situations. E XAMPLE 8.1.1 Suppose X = Y and that the C 1 -function F has the form F (λ, x) = x − G(λ, x), where G(λ, ·) : X → Y is a compact nonlinear operator with G(λ, 0) = 0 for all λ ∈ F. Now ∂x F [(λ0 , 0)] = I − ∂x G[(λ0 , 0)], where ∂x G[(λ0 , 0)] is a compact linear operator on X by Lemma 3.1.12. Therefore, by the Fredholm alternative 2.7.5, ∂x F [(λ0 , 0)] is a homeomorphism if and only if ker ∂x F [(λ0 , 0)] = {0} and a necessary condition for (λ0 , 0) to be a bifurcation point is that the linear equation ∂x F [(λ0 , 0)]x = 0 should have at least one nontrivial solution.

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More generally, when ∂x F [(λ0 , 0)] is Fredholm with index zero (see §2.7), {0} =  ker ∂x F [(λ0 , 0)] : X → Y is a necessary condition for (λ0 , 0) to be a bifurcation point. The following example shows that this condition is not sufficient. E XAMPLE 8.1.2 Let X = Y = C, regarded as Banach spaces over R, and let F (λ, z) = z − λz − i|z|2 z. Then ∂z F [(λ0 , 0)]z = (1 − λ0 )z and hence λ0 = 1 satisfies the condition which is necessary for (λ0 , 0) to be a bifurcation point. However, F (λ, z) = 0 ∈ C implies that (1 − λ)|z|2 = i|z|4 , and hence there are no non-trivial solutions (λ, z) ∈ R × C of the equation F (λ, z) = 0. In particular, there are no bifurcation points. In this example, ker ∂z F [(λ0 , 0)] is two-dimensional over R.

8.2 LYAPUNOV-SCHMIDT REDUCTION The Lyapunov-Schmidt procedure is a method for reducing the question of existence of solutions to an infinite-dimensional equation, locally in a neighbourhood of a known solution, to an equivalent one involving an equation in finite dimensions, quite commonly (though not always) in just two dimensions. Our setting is, as usual, the Banach spaces X and Y with a mapping F ∈ C k (U, Y ) for some k ∈ N, where U is open in F × X. T HEOREM 8.2.1

(Lyapunov-Schmidt Reduction) Suppose

F (λ0 , x0 ) = 0 ∈ Y where (λ0 , x0 ) ∈ U, the partial Fr´echet derivative L = ∂x F [(λ0 , x0 )] : X → Y is a Fredholm operator, ker(L) = {0} and q ∈ N is the codimension of range (L). Then there exist two open sets, U0 ⊂ U and V ⊂ F × ker(L), and two mappings, ψ ∈ C k (V, X) and h ∈ C k (V, Fq ), such that (λ0 , x0 ) ∈ U0 , (λ0 , 0) ∈ V and ψ(λ0 , 0) = x0 with F (λ, x) = 0 and (λ, x) ∈ U0 if and only if ψ(λ, ξ) = x for some (λ, ξ) ∈ V with h(λ, ξ) = 0. R EMARKS 8.2.2 The infinite-dimensional problem F (λ, x) = 0 is “reduced” to the equivalent finite-dimensional problem “find (λ, ξ) ∈ V ⊂ F × ker(L) such that h(λ, ξ) = 0.” In the event that F is F-analytic it will be clear from the proof and the analytic implicit function theorem 4.5.4 that h and ψ will also be F-analytic. Proof. Since L is a Fredholm operator there exists a finite-dimensional (and therefore closed) subspace Z ⊂ Y and a closed subspace W ⊂ X such that X = ker(L) ⊕ W and Y = Z ⊕ range (L) with dimension Z = q (the codimension of the range of L). Hence there exists (see §2.6) a (bounded) projection P : Y → Y such that ker P is the range of L and

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the range of P is Z. In particular, (I − P )Lx = Lx for all x ∈ X. Since X = ker(L) ⊕ W it follows that (I − P )L is a bijection, and hence a homeomorphism, from W onto range (L). For ξ ∈ ker(L), η ∈ W and λ ∈ F such that (λ, x0 + ξ + η) ∈ U , let G(λ, ξ, η) = (I − P )F (λ, x0 + ξ + η). Note that G(λ0 , 0, 0) = (I − P )F (λ0 , x0 ) = 0 and ∂η G[(λ0 , 0, 0)]η = (I − P )∂x F [(λ0 , x0 )]η = (I − P )Lη for all η ∈ W. Hence ∂η G[(λ0 , 0, 0)], is a homeomorphism from W onto the range of L and, by the implicit function theorem 3.5.4 (4.5.4 when F is analytic), there exist open sets U0 ⊂ U , V ⊂ F × ker(L), a mapping φ ∈ C k (V, W ) such that (λ0 , 0) ∈ V, (λ0 , x0 ) ∈ U0 , φ(λ0 , 0) = 0 and G(λ, ξ, φ(λ, ξ)) = 0 for all (λ, ξ) ∈ V, and such that {(λ, x0 + ξ + η) ∈ U0 : (I − P )F (λ, x0 + ξ + η) = 0} = {(λ, x0 + ξ + η) : (λ, ξ) ∈ V and η = φ(λ, ξ)}. (8.1) It therefore suffices to put ψ(λ, ξ) = x0 + ξ + φ(λ, ξ) and h(λ, ξ) = P F (λ, ψ(λ, ξ)) ∈ Z.

(8.2)

Then for all (λ, ξ) ∈ V , h(λ, ξ) = 0 if and only if P F (λ, x0 + ξ + φ(λ, ξ)) = 0 if and only if F (λ, x0 + ξ + φ(λ, ξ)) = 0, because (I − P )F (λ, x0 + ξ + φ(λ, ξ)) = 0. Finally choose a basis for the qdimensional space Z and thereby identify Z with Fq . R EMARKS 8.2.3 In what follows we use the notation in (8.1) and (8.2), where P is the projection from Z ⊕ range (L) onto Z ≈ Fq .

8.3 CRANDALL-RABINOWITZ TRANSVERSALITY In this section we present an important condition sufficient to guarantee that (λ0 , 0) is a bifurcation point on the line of trivial solutions of F (λ, x) = 0. The following theory of bifurcation is due to Crandall and Rabinowitz [22] who were among the first to analyse such questions systematically using implicit function theorems.

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T HEOREM 8.3.1 Suppose that X and Y are Banach spaces, that F : F × X → Y is of class C k , k ≥ 2, and that F (λ, 0) = 0 ∈ Y for all λ ∈ F. Suppose also that L = ∂x F [(λ0 , 0)] is a Fredholm operator of index zero; ker(L) is one-dimensional; ker(L) = {ξ ∈ X : ξ = sξ0 for some s ∈ F}, ξ0 ∈ X \ {0}; the transversality condition holds: 2 F [(λ0 , 0)](1, ξ0 ) ∈ / range (L). ∂λ,x

(8.3)

Then (λ0 , 0) is a bifurcation point. More precisely, there exists > 0 and a branch of solutions {(λ, x) = (Λ(s), sχ(s)) : s ∈ F, |s| < } ⊂ F × X, such that Λ(0) = λ0 ; χ(0) = ξ0 ; F (Λ(s), sχ(s)) = 0 for all s with |s| < ; Λ and s → sχ(s) are of class C k−1 , and χ is of class C k−2 , on (− , ); there exists an open set U0 ⊂ F × X such that (λ0 , 0) ∈ U0 and {(λ, x) ∈ U0 : F (λ, x) = 0, x = 0} = {(Λ(s), sχ(s)) : 0 < |s| < }; if F is analytic, χ and Λ are analytic functions on (− , ). R EMARKS 8.3.2 χ is a function from (− , ) → X. The notation in (8.3) is defined in Remark 3.2.4 where ∂λ,x F [(λ0 , 0)](1, ξ0 ) = lim

t→0

∂x F [(λ0 + t, 0)]ξ0 − ∂x F [(λ0 , 0)]ξ0 ∈ Y. t

Proof. Let U0 , V, φ, ψ, and h be given by Lyapunov-Schmidt reduction of the equation F (λ, x) = 0 in a neighbourhood of the point (λ0 , 0) ∈ F × X. Note that (8.1) with x0 = 0 implies that φ(λ, 0) = 0, since F (λ, 0) = 0, for all λ ∈ R. It is now easily confirmed that h(λ, 0) = P F (λ, 0 + φ(λ, 0)) = 0 for all (λ, 0) ∈ V ; ∂ξ h[(λ0 , 0)] = P L(Iξ + ∂ξ φ[(λ, 0)]) = 0 (Iξ is the identity on ker(L)); 2 2 h[(λ0 , 0)](1, ξ0 ) = P ∂λ,x F [(λ0 , 0)](1, ξ0 ) = 0; ∂λ,ξ

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h ∈ C k (V, F), k ≥ 2, where the one-dimensional space Z has, as in §8.2, been identified with F and V ⊂ F×ker(L) is an open set containing (λ0 , 0). Now define g : V → F by


1

∂ξ h[(λ, tξ)] ξ0 dt.

g(λ, ξ) = 0

It is immediate that g is of class C k−1 (when F (and therefore h) is analytic, g is analytic) and that 2 g(λ0 , 0) = 0, ∂λ g(λ0 , 0) = ∂λ,ξ h[(λ0 , 0)](1, ξ0 ) = 0.

The implicit function theorem 3.5.4 now gives the existence of a mapping Λ ∈   C k−1 {s ∈ F : |s| < }, F for some > 0, such that Λ(0) = λ0 and g(Λ(s), sξ0 ) = 0 if |s| < . To complete the proof observe, since h(λ, 0) = 0, that  h(λ, ξ)   if s = 0, s for (λ, ξ) = (λ, sξ0 ) ∈ V. g(λ, ξ) =   ∂ξ h(λ, 0)ξ0 if s = 0, Now put χ(s) = s−1 ψ(Λ(s), sξ0 ) for 0 < |s| < and χ(0) = ξ0 . In fact lim χ(s) = ∂λ ψ[(λ0 , 0)]Λ
(0) + ∂ξ ψ[(λ0 , 0)]ξ0 = ξ0 ,

0 =s→0

and so χ is continuous at s = 0. Indeed, for all s with |s| < ,
1 χ(s) = ∂λ ψ[(Λ(ts), tsξ0 )]Λ
(ts) + ∂ξ ψ[(Λ(ts), tsξ0 )]ξ0 dt, 0

from which it follows easily that χ is of class C k−2 , and F-analytic in the case that F , and consequently Λ and ψ, are F-analytic. This completes the proof. E XAMPLES 8.3.3 (Concerning Transversality) (a) Let F = R, X = Y = R and let F (λ, x) = x(λ2 + x2 ). Clearly F : R × R → R is C 2 , F (λ, 0) = 0 for all λ ∈ R, L = ∂x F [(0, 0)] is the zero operator, ker(L) = X = R and R(L) = {0}, 2 F [(0, 0)] is the zero but the transversality condition is not satisfied because ∂λ,x 2 element of M (R; R). It is easily seen that (0, 0) ∈ R × X is not a bifurcation point. (b) If, in example (a), F (λ, x) = x(λ + x2 ), then the transversality condition holds and (of course) (0, 0) is a bifurcation point. (c) If however F (λ, x) = x(λ3 + x2 ), then (0, 0) is a bifurcation point although the transversality condition fails.

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P ROPOSITION 8.3.4 (a) Let the hypotheses of Theorem 8.2.1 be satisfied and let U0 , V, h and ψ be given by its conclusion. Then U0 and V can be chosen sufficiently small that for all (λ, ξ) ∈ V     dim ker ∂x F [(λ, ψ(λ, ξ))] = dim ker(∂ξ h[(λ, ξ)] . (b) Suppose that the hypotheses of Theorem 8.3.1 are satisfied and that, in part (a) above, U = F × X and (λ0 , x0 ) = (λ0 , 0). Now let Λ, χ, be as in the conclusion of Theorem 8.3.1. Then   dim ker ∂x F [(Λ(s), sχ(s))] ∈ {0, 1}   and, for s with 0 < |s| < , dim ker ∂x F [(Λ(s), sχ(s))] = 1 if and only if Λ
(s) = 0 . Proof. From the identity (I − P )F (λ, x0 + ξ + φ(λ, ξ)) ≡ 0, (λ, ξ) ∈ V, it follows that (I − P )∂x F [(λ, x0 + ξ + φ(λ, ξ))](v + ∂ξ φ[(λ, ξ)]v) ≡ 0 for all v ∈ ker(L). Therefore, if ∂x F [(λ, x0 + ξ + φ(λ, ξ))](v + w) = 0 with v ∈ ker(L) and w ∈ W , then w − ∂ξ φ[(λ, ξ)]v ∈ W and (I − P )∂x F [(λ, x0 + ξ + φ(λ, ξ))](w − ∂ξ φ[(λ, ξ)]v) = 0.

 Since (I −P )L : W → range (L) is bijective, (I −P )∂x F [(λ, x0 +ξ +φ(λ, ξ))]W is also a bijection (in particular it is injective) when the sets U0 and V have been chosen with sufficiently small diameters (see §2.5). Therefore w = ∂ξ φ[(λ, ξ)]v and so ∂x F [(λ, x0 + ξ + φ(λ, ξ))](v + w) = 0 for some w ∈ W if and only if P ∂x F [(λ, x0 + ξ + φ(λ, ξ))](v + ∂ξ φ[(λ, ξ)]v) = 0 if and only if ∂ξ h[(λ, ξ)]v = 0.   Part (a) follows. In part (b), dim ker ∂x F [(Λ(s), sχ(s))] ∈ {0, 1} because ∂ξ h[(λ, ξ)] maps the one-dimensional space span ξ0 into itself. For the second part of (b) note that, for s with |s| < , d h(Λ(s), sξ0 ) = ∂λ h[(Λ(s), sξ0 )]Λ
(s) + ∂ξ h[(Λ(s), sξ0 )]ξ0 ds = s∂λ g[(Λ(s), sξ0 )]Λ
(s) + ∂ξ h[(Λ(s), sξ0 )]ξ0

0=

with ∂λ g[(Λ(s), sξ0 )] = 0 if > 0 is sufficiently small. (Recall the hypothesis that 2 2 ∂λ g[(λ0 , 0)](1) = ∂λ,ξ h[(λ0 , 0)](1, ξ0 ) = P ∂λ,x F [(λ0 , 0)](1, ξ0 ) = 0.)

This proves that, for any s with 0 < |s| < ,   dim ker ∂x F [(Λ(s), sχ(s))] = 1 if and only if Λ
(s) = 0.

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8.4 BIFURCATION FROM A SIMPLE EIGENVALUE Here is a case where transversality is trivial to verify. Recall the definition of a simple eigenvalue 2.7.8 T HEOREM 8.4.1 (Bifurcation from a Simple Eigenvalue) Suppose that the real Banach space X is continuously embedded in the real Banach space Y and that {(λ, 0) : λ ∈ R} ⊂ U ⊂ R × X, where U is open. Suppose that F ∈ C k (U, Y ), k ≥ 2 and for all λ ∈ R, F (λ, 0) = 0 and ∂x F [(λ, 0)] = λ ι − A,

(8.4)

where ι is the continuous embedding of X in Y . Then every simple eigenvalue λ0 of A is a bifurcation point and the conclusion of Theorem 8.3.1 holds. Proof. Because of the hypotheses, the transversality condition (8.3) demands that λ0 ι − A be Fredholm of index 0 with one-dimensional kernel spanned by ξ0 say, and that 2 ξ0 = ∂λ,x F [(λ0 , 0)](1, ξ0 ) ∈ / range (λ0 ι − A).

However, this is precisely the requirement that λ0 is a simple eigenvalue of A. The result now follows as a special case of Theorem 8.3.1. Suppose the hypotheses of Theorem 8.4.1 hold. Let y ∗ ∈ Y ∗ be such that y (ξ0 ) = ξ0 , y ∗ = 1, where y ∗ (range (λ0 ι − A)) = 0 and ξ0 spans the kernel of λ0 ι − A, as in the proof of Proposition 3.6.1. Let (Λ(s), sχ(s)) be as in the conclusion of Theorem 8.4.1 and let ∗

L(s) = ∂x F [(Λ(s), sχ(s))] ∈ L(X, Y ), s ∈ (− , ), L(0) = λ0 ι − A. Then, by Theorem 3.6.1, there is a C k−1 -curve {(µ(s), ξ(s)) : s ∈ (− , )} ⊂ R × X such that (µ(0), ξ(0)) = (0, ξ0 ), L(s)ξ(s) = µ(s)ιξ(s) and y ∗ (ιξ(s)) = 1, s ∈ (− , ), where ξ(s) = ξ0 +η(s), η is of class C k−1 , η(0) = 0 and ιη(s) ∈ X ∩range (L(0)). The next result explains the direction in which µ(s) moves as s passes through 0, an observation which is important when deciding the stability and the Morse index of the bifurcating solutions, see §11.3. P ROPOSITION 8.4.2

In this notation, lim

s→0 sΛ
(s) =0

µ(s) = −1. sΛ
(s)

Proof. For s ∈ (− , ), in the notation of (8.2) let x(s) = sχ(s) = ψ(Λ(s), sξ0 ) = sξ0 + φ(Λ(s), sξ0 ) = sξ0 + ρ(s), say.

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(This is the definition of ρ.) Since F (Λ(s), x(s)) = 0, differentiation gives ∂λ F [(Λ(s), x(s)]Λ
(s) + L(s)x
(s) = 0 for all s ∈ (− , ). Therefore L(s)ξ(s) = µ(s)ι ξ(s), s ∈ (− , ), implies that   L(s) ξ(s) − x
(s) = ∂λ F [(Λ(s), x(s))]Λ
(s) + µ(s)ιξ(s). 2 Since ∂λ22 F [(λ0 , 0)] = 0 and ∂λ,x F [(λ0 , 0)](λ, x) = λx, we find that

∂λ F [(Λ(s), x(s))] = ∂λ22 F [(λ0 , 0)](Λ(s) − λ0 ) 2 + ∂λ,x F [(λ0 , 0)]x(s) + o(|Λ(s) − λ0 |) + o( x(s) )

= sιξ0 + ιρ(s) + o(|Λ(s) − λ0 |) + o( x(s) ) = sξ0 + o(|s|) as s → 0. Therefore, since ξ(s) − x
(s) = η(s) − ρ
(s),   L(s) η(s) − ρ
(s) = Λ
(s)(sιξ0 + o(|s|)) + µ(s)ι(ξ0 + η(s)), and so L(0)(η(s) − ρ
(s)) = Λ
(s)(sιξ0 + o(|s|)) + µ(s)ι(ξ0 + η(s))   + L(0) − L(s) (η(s) − ρ
(s)). (8.5) By Lemma 2.7.9, X1 = X ∩ range (L(0)) is closed in X and L(0) = λ0 ι − A is a homeomorphism from X1 onto range (L(0)) ⊂ Y . Since η(s) − ρ
(s) ∈ X1  

η(s) − ρ
(s) X ≤ const. |sΛ
(s)| + |µ(s)| , s ∈ (− , ). Applying y ∗ to both sides of (8.5) gives     − sΛ
(s) + µ(s) ξ0 = o(|s|)Λ
(s) + y ∗ (L(0) − L(s))(η(s) − ρ
(s)) . Therefore, for > 0 sufficiently small, |sΛ
(s) + µ(s)| ≤ o(|s|)|Λ
(s)| + o(1)|µ(s)|, s ∈ (− , ).

(8.6)

Since η is continuous and η(0) = 0, |µ(s)| ≤ |sΛ
(s)| + |sΛ
(s) + µ(s)| ≤ |sΛ
(s)| + o(|s|) |Λ
(s)| + o(1)|µ(s)| as s → 0. Therefore |µ(s)| ≤ const. |sΛ
(s)|, s ∈ (− , ). Similarly, |sΛ
(s)| ≤ const. |µ(s)|. Therefore (8.6) implies that |sΛ
(s) + µ(s)| = o(|sΛ
(s)|) and |sΛ
(s) + µ(s)| = o(|µ(s)|) as s → 0. The result now follows.

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LOCAL BIFURCATION THEORY

We now give two applications of the local-bifurcation method just explained. The first was featured in the Introduction where a global theory was developed by solving the equation more-or-less explicitly. Here we see that the bifurcating of solutions observed there is a consequence of general abstract considerations and the theory of bifurcation from a simple eigenvalue.

8.5 BENDING AN ELASTIC ROD II We now show how local bifurcation theory applies to the boundary-value problem (1.3): φ

(x) + λ sin φ(x) = 0 for x ∈ [0, L], φ
(0) = φ
(L) = 0,

(8.7)

where L is fixed and λ > 0 is the parameter in the problem. Let F = R, X = {φ ∈ C 2 [0, L] : φ
(0) = φ
(L) = 0}, Y = C[0, L], and define F (λ, φ) = φ

+ λ sin φ. It is not difficult to see (as in Example 4.3.6) that F : R × X → Y is R-analytic. Moreover ∂φ F [(λ, 0)]φ = 0 if and only if φ

+ λφ = 0 ∈ Y and φ
(0) = φ
(L) = 0. This linear boundary-value problem has solutions (λ, φ) with λ > 0 and φ =  0 if  2 : K ∈ N}. Then φ is a multiple of φ and only if λ ∈ {λK = Kπ K where L Kπs φK (s) = cos L and dim ker ∂φ F [(λK , 0)] = 1. Moreover    range ∂φ F [(λK , 0)] = v ∈ C[0, L] :




L

 v(s)φK (s)ds = 0 .

0



To see this note that if u ∈ X and u + λK u = v ∈ Y , then an integration by parts implies that





L

L

v(s)φK (s)ds = 0

(u

(s) + λn u(s))φK (s)ds

0


=

L

(φ

K (s) + λK φ(s))u(s)ds = 0.

0

That every v ∈ Y with


L

v(s)φK (s)ds = 0 0

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is in the range of ∂φ F [(λK , 0)] can be shown using the variation of constants formula. Let A : X → Y be defined by Aφ = −φ

. Then λK , K ∈ N, is a simple eigenvalue of A. Therefore, for all K ∈ N, there is a bifurcation from a simple eigenvalue for (8.7) at every point (λK , 0) on the line of trivial solutions.

8.6 BIFURCATION OF PERIODIC SOLUTIONS This section concerns a simple example of the existence, via bifurcation from a line of trivial solutions, of non-constant but periodic, solutions  of a differential equation. Let δ > 0 and suppose that the functions A, B ∈ C 2 (−δ, δ) × R × (−δ, δ); R are 2π-periodic in the second variable, t (time). Now consider the boundary-value problem on [0, 2π] w(t) ˙ + λw(t) + w(t)2 A(w(t), t, λ) + λ2 w(t)B(w(t), t, λ) = 0, w(0) = w(2π), w ∈ C 1 [0, 2π], where w˙ = dw/dt. Note that for all λ ∈ R, w = 0 is a time-independent solution of this problem. To establish the existence of time-dependent solutions let F = R, X = {u ∈ C 1 [0, 2π] : u(0) = u(2π), u
(0) = u
(2π)}, Y = {v ∈ C[0, 2π] : v(0) = v(2π)} and, for 0 ≤ t ≤ 2π and u ∈ X, let F (λ, u)(t) = u(t) ˙ + λu(t) + u(t)2 A(u(t), t, λ) + λ2 u(t)B(u(t), t, λ). Then F ∈ C 2 (U, Y ), where U = {(λ, u) : |λ| < δ,

max |u(t)| < δ} ⊂ R × X.

0≤t≤2π

Moreover ∂u F [(0, 0)]u = u, ˙ u ∈ X,   ker ∂u F [(0, 0)] = {u ∈ X : u is a constant},   * 2π range ∂u F [(0, 0)] = {v ∈ Y : 0 v(t)dt = 0}, and therefore ∂u F [(0, 0)] is a Fredholm operator of index zero. Let ξ0 = 1. It follows that  2 ∂λ,u F [(0, 0)](1, ξ0 ) = ξ0 ∈ / range ∂u F [(0, 0)]. Therefore, by Theorem 8.3.1, there exists a C 1 -curve {(Λ(s), sχ(s)) ∈ U : s ∈ (− , )}

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LOCAL BIFURCATION THEORY

such that F (Λ(s), sχ(s)) = 0, χ(0) = ξ0 and Λ(0) = 0. For s ∈ (0, ) sufficiently small, (λ, w) = (Λ(s), sχ(s)) is a periodic solution of our problem for which w is positive (because χ(s) = 1 + η(s) and η(0) = 0 in X) on [0, 2π]. To ensure that, for > 0 sufficiently small, this is not a constant solution we need an extra hypothesis, such as ∂t A[(0, t, 0)] ≡ 0,

(8.8)

which ensures that constant solutions do not bifurcate from the line of trivial solutions at (0, 0). Suppose that (8.8) holds and that there is a sequence {(λn , cn )} of solutions, where cn = 0 is a constant, converging to (0, 0). Then λn + cn A(cn , t, λn ) + λ2n B(cn , t, λn ) = 0. After dividing by (λn , cn ) and taking the limit of a subsequence as n → ∞, we obtain a vector (λ∗ , c∗ ) with (λ∗ , c∗ ) = 1 satisfying λ∗ + c∗ A(0, t, 0) = 0. This contradicts (8.8).

8.7 NOTES ON SOURCES This material is now completely standard, see Ambrosetti and Prodi [2], Chow and Hale [19], Crandall and Rabinowitz [22, 23], Schwartz [52], Stuart [54, 55], Zeidler [68], Toland [61], and many other references.

Chapter Nine Global Bifurcation Theory Let X, Y be Banach spaces over R, let U ⊂ R × X and F : U → Y be an R-analytic function. Suppose that (G1) (λ, 0) ∈ U and F (λ, 0) = 0 for all λ ∈ R. (G2) ∂x F [(λ, x)] is a Fredholm operator of index zero when F (λ, x) = 0, (λ, x) ∈ U . (G3) For some λ0 ∈ R,

  ker ∂x F [(λ0 , 0)] = {sξ0 : s ∈ R},  2 ∂λ,x F [(λ0 , 0)](1, ξ0 ) ∈ / range ∂x F [(λ0 , 0)]).

By Theorem 8.3.1, there exists an analytic function (Λ, κ) : (− , ) → R × X such that (λ, x) = (Λ(s), κ(s)) satisfies F (λ, x) = 0 for all s ∈ (− , ), Λ(0) = λ0 and κ
(0) = ξ0 . (In the notation of §8.3, κ(s) = sχ(s).) Let R+ = {(Λ(s), κ(s)) : s ∈ (0, )}, S = {(λ, x) ∈ U : F (λ, x) = 0}, T = {(λ, x) ∈ S : x = 0}. Suppose > 0 is sufficiently small that κ
(s) = 0 for s ∈ (− , ) and R+ ⊂ T .

9.1 GLOBAL ONE-DIMENSIONAL BRANCHES The following result gives a global extension of the function (Λ, κ) from (0, ) to (0, ∞) in the R-analytic case. However, in proving it we develop a general approach to the global extendability of one-parameter curves of non-singular solutions to an equation F (λ, x) = 0. T HEOREM 9.1.1 Suppose (G1)–(G3) hold, Λ
≡ 0 on (− , ) and that in R × X all bounded closed subsets of S are compact. Then there exists a continuous curve R which extends R+ as follows. (a) R = {(Λ(s), κ(s)) : s ∈ [0, ∞)} ⊂ U where (Λ, κ) : [0, ∞) → R × X is continuous. (b) R+ ⊂ R ⊂ S.

115

GLOBAL BIFURCATION THEORY

    (c) The set s ≥ 0 : ker ∂x F [(Λ(s), κ(s))] = {0} has no accumulation points. (d) At each point, R has a local analytic re-parameterization in the following sense. In a right neighbourhood of s = 0, R and R+ coincide. For each s∗ ∈ (0, ∞) there exists ρ∗ : (−1, 1) → R which is continuous, injective, and ρ∗ (0) = s∗ , t → (Λ(ρ∗ (t)), κ(ρ∗ (t))), t ∈ (−1, 1), is analytic. Furthermore Λ is injective on a right neighbourhood of 0 and for s∗ > 0 there exists ∗ > 0 such that Λ is injective on [s∗ , s∗ + ∗ ] and on [s∗ − ∗ , s∗ ]. (e) One of the following occurs. (i) (Λ(s), κ(s)) → ∞ as s → ∞. (ii) (Λ(s), κ(s)) approaches the boundary of U as s tend to ∞. (iii) R is a closed loop. In other words, for some T > 0, R = {(Λ(s), κ(s)) : 0 ≤ s ≤ T } and (Λ(T ), κ(T )) = (λ0 , 0). We may suppose that T > 0 is the smallest such T and (λ(s + T ), κ(s + T )) = (Λ(s), κ(s)) for all s ≥ 0. (f) If, for some s1 = s2 , (Λ(s1 ), κ(s1 )) = (Λ(s2 ), κ(s2 )) where ker ∂x F [(Λ(s1 ), κ(s1 ))] = {0}, then (e)(iii) occurs and |s1 − s2 | is an integer multiple of T . In particular, (Λ, κ) : [0, ∞) → S is locally injective. R EMARKS 9.1.2 (1) There is no claim that R is a maximal connected subset of S. Other curves or manifolds in S may intersect R. (2) R may self-intersect in the sense that while s → (Λ(s), κ(s)) is locally injective, it need not be globally injective. For example, in part (e)(iii) of the theorem it is clearly not globally injective. (3) In part (d) it can happen that the parametrization has zero derivative, in which case {(Λ(s), κ(s)) : |s − s∗ | < δ ∗ } ⊂ R may not be a smooth curve even though it has a local analytic parameterization at every point. Of course, for δ ∗ sufficiently small, the two segments of the set {(Λ(s), κ(s)) : 0 < |s − s∗ | < δ ∗ }, with (Λ(s∗ ), κ(s∗ )) deleted, are smooth and can be parameterized by λ. (4) Alternative (e)(i) is much stronger than the claim that R is unbounded in R × X.

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Proof. The proof of Theorem 9.1.1 is organized below in a few short steps. Let   N = {(λ, x) ∈ S : ker ∂x F [(λ, x)] = {0}} (the non-singular solutions of F (λ, x) = 0). By hypothesis Λ ≡ 0 on (− , ) and Λ is R-analytic. Therefore Λ
is nowhere zero on (− , ) \ {0} for > 0 sufficiently small and, by Proposition 8.3.4(b), we may assume that > 0 is such that R+ ⊂ N. D EFINITION 9.1.3 nected subset of N.

(9.1)

(Distinguished Arcs) A distinguished arc is a maximal con-

Hypothesis G(2) and the analytic implicit function Theorem 4.5.4 ensure that a distinguished arc I is the graph of an R-analytic function of λ. More precisely, if I is a distinguished arc then there exists a (possibly infinite) open interval I and an R-analytic function g : I → X such that {(λ, g(λ)) : λ ∈ I} = I.

(9.2)

Step 1. (Lyapunov-Schmidt Reduction) We need to connect the theory of varieties with the present infinite-dimensional problem. To study the structure of S in a neighbourhood of a point (λ∗ , x∗ ) ∈ S \ N we use the Lyapunov-Schmidt procedure. Theorem 8.2.1 yields the existence of a neighbourhood V of (λ∗ , 0) in R × ker(∂x F [(λ∗ , x∗ )]), R-analytic maps ψ : V → X, h : V → Rq (q = dim ker(∂x F [(λ∗ , x∗ )])) with (a) ψ(λ∗ , 0) = x∗ and (λ, ψ(λ, ξ)) ∈ U for (λ, ξ) ∈ V . (b) For all (λ, ξ) ∈ V , h(λ, ξ) = 0 if and only if F (λ, ψ(λ, ξ)) = 0. (c) If F (λ, x) = 0, (λ, x) ∈ U and (λ, x)−(λ∗ , x∗ ) is sufficiently small, then there exists ξ ∈ ker(∂x F [(λ∗ , x∗ )]) such that (λ, ξ) ∈ V and x = ψ(λ, ξ). (d) dim ker(∂x F [(λ, ψ(λ, ξ))]) = dim ker(∂ξ h[(λ, ξ)]), (λ, ξ) ∈ V. Recall the notation of §7. The analytic function h : V → Rq may be identified with the set of its q component functions each of which maps V into R analytically. Therefore we may define an R-analytic variety A and a manifold M by A = var (V, {h}) = {(λ, ξ) ∈ V : h(λ, ξ) = 0}, M = {(λ, ξ) ∈ V : (λ, ψ(λ, ξ)) ∈ N}. By Proposition 8.3.4(a), the elements of M are 1-regular points of A. Let {Mj : j ∈ J} denote those non-empty connected components of M which have the property that γ(λ∗ ,0) (Mj ) = ∅. Since h is an R-analytic function on the (q +1)-dimensional real vector space V , the q components of h(λ, ξ) are real functions defined locally in a neighbourhood

117

GLOBAL BIFURCATION THEORY

of (λ∗ , 0) ∈ V by a Taylor series, the nth term of which is a sum of terms of the form q+1 h∗k1 ,··· ,kq+1 xk11 · · · xq+1 ,

k

where k1 + · · · + kq+1 = n and h∗k1 ,··· ,kq+1 ∈ R. Here (x1 , · · · xq+1 ) ∈ Rq+1 are the coefficients of (λ∗ , 0)−(λ, ξ) in some linear coordinate system. Replacing (x1 , · · · , xq+1 ) ∈ Rq+1 with (z1 , · · · , zq+1 ) ∈ Cq+1 leads to a real-on-real C-analytic extension hc of h in a complex neighbourhood V c of (λ∗ , 0) and a corresponding C-analytic variety. Let Ac = var (V c , {hc }) = {(λ, ξ) ∈ V c : hc (λ, ξ) = 0},   M c = (λ, ξ) ∈ V c : ker(∂ξ hc [(λ, ξ)]) = {0} , and let {Mjc : j ∈ J c } be the non-empty connected components of M c with γ(λ∗ ,0) (Rq+1 ∩ Mjc ) = ∅. Note that for each j ∈ J there exists ˆj ∈ J c such that Mj ⊂ Mˆjc . Step 2. (Application of the Structure Theorem) Theorem 7.4.7 (d)–(f) on the structure of complex analytic varieties, when applied to Ac gives, for each j ∈ J c , the existence of a real-on-real branch Bj with γ(λ∗ ,0) (Mjc ) ⊂ γ(λ∗ ,0) (B j ), dim Bj = 1 and Bj ⊂ Ac . By making the neighbourhood V c smaller if necessary, we may suppose that Bj \ {(λ∗ , 0)} ⊂ Mjc . By Theorem 7.4.7 there are finitely many branches and hence finitely many Mjc and Mj . By Theorem 7.5.1 each of these one-dimensional branches Bj admits a C-analytic parameterization in a neighbourhood of (λ∗ , 0). We now return to the setting of Rn . From Corollary 7.5.3, we obtain that M , locally near (λ∗ , 0), is the union of a finite number of curves which pass through (λ∗ , 0) in V , intersect one another only at (λ∗ , 0) and are given by the parameterization (7.16). Thus, in our previous notation, each Mj , j ∈ J, is paired, in a unique way with another M˜j , ˜j ∈ J, so that their union with the point (λ∗ , 0) forms one of these curves in V . This observation can be lifted to infinite dimensions as follows. Suppose that I in (9.2) is a distinguished arc where I = (a, b) with (b, g(b)) ∈ S \ N. Then the germ γ(λ∗ ,0) (I) coincides with the germ of the image under the mapping (λ, ξ) → (λ, ψ(λ, ξ)) of Mj for some j ∈ J. Hence I has a unique extension beyond (b, g(b)) given by the image of M˜j under the same mapping. D EFINITION 9.1.4 (Routes of Length N ) A route of length N ∈ N ∪ {∞} is a set {An : 0 ≤ n < N } of distinguished arcs and a set {(λn , xn ) : 0 ≤ n < N } ⊂ R × X such that (a) (λ0 , x0 ) = (λ0 , 0) is the bifurcation point; (b) R+ ⊂ A0 ;

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CHAPTER 9

(c) For N > 1 and 0 ≤ n < N − 1,   (λn+1 , xn+1 ) ∈ ∂An ∩ ∂An+1 \ {(λn , xn )} and there exists an injective R-analytic map ρ : (−1, 1) → An ∪ An+1 ∪ {(λn+1 , xn+1 )} with ρ(0) = (λn+1 , xn+1 ). Hence An+1 is uniquely determined by An and vice versa. (d) The mapping n → An is injective. Step 3. (Existence of a Maximal Route) That {A0 }, {(λ0 , 0)} is a route of length 1 with (λ0 , 0) ∈ ∂A0 is obvious from the discussion leading to (9.1). Parts (c) and (d) of the definition of a route imply that An+1 = An and that An+1 is uniquely determined by An . Therefore if {Ajn , 0 ≤ n < Nj }, {(λjn , xjn ) : 0 ≤ n < Nj }, j ∈ {1, 2}, are two routes with N1 ≤ N2 it follows that λ1n = λ2n , x1n = x2n for all n with 0 ≤ n < N1 . Hence, under the hypotheses of Theorem 9.1.1, there exists a maximal route of length N ∈ N ∪ {∞} which we denote by {An , (λn , xn )} : 0 ≤ n < N }. Step 4. (Parameterization of a Maximal Route) Because of the remark following Definition 9.1.3, An = {(Λn (s), κn (s)), s ∈ (n, n + 1)}, 0 ≤ n < N, where (Λn , κn ) is an R-analytic function. There are three cases: N = ∞; N < ∞ when AN −1 is not a compact subset of U ; and N < ∞ when AN −1 is a compact subset of U . Suppose that An is a bounded closed subset of U for some 0 ≤ n < N −1. Then the parameterization of An by s ∈ (n, n+1) can be extended as a parameterization of An by s ∈ [n, n + 1] with (Λn (m), κn (m)) = (λm , xm ) when m = n and m = n + 1. Since n < N − 1, Definition 9.1.4 implies that An+1 can be parameterized, in a neighbourhood of (λn , xn ) by s ∈ [n + 1, n + 1 + ), for some > 0, and so, in the first two cases above, (Λ(n), κ(n)) = (λn , xn ), n = 0, 1, · · · , N − 1, (Λ(s), κ(s)) = (Λn (s), κn (s)) for s ∈ (n, n + 1), defines a continuous parameterization {(Λ(s), κ(s)) : 0 ≤ s < N }

(9.3)

119

GLOBAL BIFURCATION THEORY

of   R = ∪0≤n 0 and (e)(iii) in Theorem 9.1.1 does not occur. Proof. Let s = sup{s > 0 : κ((0, s]) ⊂ K \ {0}} and suppose, for contradiction, that s < ∞. Since K is closed, κ(s) ∈ K. Moreover κ(s) = 0, since otherwise κ(s) ∈ K for some s > s by hypothesis (d) of the present theorem. Therefore (Λ(s), 0) is a bifurcation point on the line of trivial solutions of the equation F (λ, x) = 0. Since (Λ(s), κ(s)) ∈ R, we can assume that, in a neighbourhood of s, R is parameterized R-analytically by s. Let k denote the smallest natural number such that the k th derivative of κ at s = s is non-zero (k exists, by analyticity), so that κ(s) = κ(s) − κ(s) =

dk κ[s] (s − s)k + O(|s − s|k+1 ). k!

Since, by definition of s, κ(s) ∈ K for all s with 0 ≤ s < s, we conclude that (−1)k dk κ[s] ∈ K \ {0}.

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GLOBAL BIFURCATION THEORY

Note that ∂λmm F [(λ, 0)] = 0 for all m. So differentiating the identity F (Λ(s), κ(s)) = 0 k times at s = s leads to the conclusion that     (−1)k dk κ[s] ∈ ker ∂x F [(Λ(s), κ(s))] ∩ K \ {0} . Now hypothesis (c) implies that Λ(s) = λ0 and (−1)k dk κ[s] is a positive multiple of ξ0 . Since λ0 is a simple eigenvalue and −ξ0 ∈ K, the bifurcating curve which lies in R×K and passes through the bifurcation point (λ0 , 0) is uniquely determined as R+ . Thus there is a segment of R, parameterized by s < s sufficiently close to s, which is a subset of R+ . Since R+ ⊂ A0 , there exist sequences {sk }, {tk } such that (Λ(sk ), κ(sk )) = (Λ(s − tk ), κ(s − tk )), sk  0, tk  0. By Theorem 9.1.1 (f), T > 0 divides s − tk − sk for all k, which is false. Hence s = ∞ and κ(s) ∈ K \ {0} for all s > 0. Since (λ0 , 0) ∈ R we conclude that R does not form a loop and the proof is complete.

9.3 BENDING AN ELASTIC ROD III We now show how the theory of global bifurcation in cones can be applied to the boundary-value problem in §8.5: φ

(x) + λ sin φ(x) = 0 for x ∈ [0, L], φ
(0) = φ
(L) = 0,

(9.5)

where L is fixed and λ > 0 is the parameter in the problem. As before let F = R, X = {φ ∈ C 2 [0, L] : φ
(0) = φ
(L) = 0}, Y = C[0, L], and define F (λ, φ) = φ

+ λ sin φ. Then F : R × X → Y is R-analytic, ∂φ F [(λ, 0)]φ = 0 if and only if φ

+ λφ = 0 ∈ Y and φ
(0) = φ
(L) = 0 and the bifurcation points form the set {λK = (Kπ/L)2 : K ∈ N}. Here we focus on finding a global extension of the local bifurcation at the point (π/L)2 corresponding to K = 1. In keeping with the notation of the last section let λ0 denote (π/L)2 and let ξ0 (x) = cos(πx/L), x ∈ [0, L]. Next we verify the hypotheses of Theorem 9.2.2. We have already seen that (G1) and (G3) hold. To check (G2) let (λ, ψ) ∈ R × X be a solution of (9.5). Then dφ F [(λ, φ)](ψ) = ψ

+ λψ cos φ, ψ ∈ X.

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By the theory of ordinary differential equations, ψ

+λψ cos φ = 0 has two linearly independent solutions at most one of which is in X. If there are no solutions in X the problem ψ

+ λψ cos φ = f, ψ ∈ X

(9.6)

has a solution ψ for every f ∈ Y . If, on the other hand, it has a solution ψˆ ∈ X, then (9.6) has a solution if and only if
L ˆ ψ(x)f (x) dx = 0. 0

In both cases the range is closed, the codimension of the range and the dimension of kernel of dφ F [(λ, φ)] coincide. This shows that in all cases dφ F [(λ, φ)] is a Fredholm operator of index zero and so (G2) holds. Now let K ⊂ X be the cone defined by   K = u ∈ X : u is odd about L/2 and u ≥ 0 on [0, L/2] . We have seen that in this example hypothesis (a) of Theorem 9.2.2 holds, and (c) is obvious since the (unique up to normalization) eigenfunction corresponding the eigenvalue (πK/L)2 is cos(Kπx/L) and only when  K = 1 is it in K. To see that (d) holds suppose that (λ, φ) ∈ R × K \ {0} satisfies (9.5). Then clearly λ = 0, sin φ(0) = 0 and sin φ(L) = 0. (If any one of them is zero then φ is a constant, by the uniqueness theorem for the initial-value problems for second order ordinary differential equations, and so φ is not odd about L/2.) Also any ˆ φ) ˆ of (9.5) satisfies solution (λ, 1 2

2 ˆ cos φ(0) ˆ −λ ˆ cos φ(x) ˆ ≡ 0 on [0, L] φˆ
(x) + λ

(9.7)

ˆ = 0, cos φ(0) ˆ ˆ and, if λ = cos φ(L). Since λ = 0 and the derivative of cosine at φ(0) and at φ(L) is not zero, it ˆ φ) ˆ is a solution of (9.5) which is sufficiently close to (λ, φ) then follows that if (λ, ˆ ˆ ˆ ˆ − x) solve the same initial φ(0) = −φ(L). Hence the functions φ(x) and −φ(L ˆ value problem, and so are equal. This shows that φ is odd about L/2. Now to show that φˆ ≥ 0 on [0, L/2] suppose that there is a sequence (λk , φk ) of solutions of (9.5) which converges to (λ, φ) in R × X such that φ(xk ) < 0, xk ∈ [0, L/2). Since φk (L/2) = 0 and φk (0) > 0 for k sufficiently large, we may assume that xk is a minimizer of φk on [0, L/2] and hence φ
k (xk ) = 0. In the limit as k → ∞ we find that there exists x ∈ [0, L/2] with φ(x) = φ
(x) = 0. By the uniqueness theorem for initial value problems this means that φ ≡ 0, which is false. This contradiction establishes (d). It remains to show (b), that R+ ⊂ R × K. First we show that if (λ, φ) ∈ R+ , for > 0 sufficiently small, then φ is odd about L/2. Recall from Theorem 8.4.1 that    R+ = Λ(s), s(ξ0 + τ (s) : s ∈ (0, ) ,

GLOBAL BIFURCATION THEORY

123

where Λ(s) → 1 and τ (s) → 0 in X as s → 0. To complete the proof that hypothesis (b) is satisfied recall that ξ0 (x) = cos(πx/L) and hence ξ0 (0) = 1 = −ξ0 (L). So (9.7) gives       cos s(1 + τ (s)(0)) = cos s(−1 + τ (s)(L)) = cos s(1 − τ (s)(L)) whence s(1 + τ (s)(0)) = ±s(1 − τ (s)(L)) for s > 0 sufficiently small. It follows that the sign must be plus, and τ (s)(0) = −τ (s)(L). Thus s(ξ0 + τ (s)) is odd about L/2 for s > 0 sufficiently small. Now κ(s) = s(ξ0 + τ (s)) ≥ 0 on [0, L/2] follows since κ(L/2) = 0, κ(s)
(L/2) = s(−π/L + τ (s)
(L/2)) and τ (s) → 0 in X as s → 0. Hence hypothesis (b) is satisfied. Thus Theorem 9.2.2 gives the existence of a curve   R = (Λ(s), κ(s) : s ∈ [0, ∞)   with (Λ(0), κ(0)) = (π/L)2 , 0 , κ(s) ∈ K for s > 0 and

(Λ(s), κ(s)) → ∞ as s → ∞. If now (λ, φ) = (Λ(s), κ(s)) ∈ R satisfies (9.5) it is obvious that λ = 0 and, by connectedness, λ > 0 for all (λ, φ) ∈ R. Multiplying (9.5) by ξ0 and integrating by parts gives
L
L

π   λ sin φ  2 dx ξ0 φ

+ sin φ dx = φ ξ0 − + 0= L φ 0 0 Since φ, ξ0 ∈ K, the product φξ0 is non-negative and not identically zero. Since λ > 0 and (λ sin φ)/φ < λ, it follows that λ > (π/L)2 for all (λ, φ) ∈ R, φ = 0. Hence the global curve lies to the right of the bifurcation point. Since φ
(0) = 0 = φ(L/2) for all solutions of (9.5), it is immediate that the set   (λ, φ) ∈ R : λ ≤ M is bounded in R × X for all finite M . Since R is unbounded,   λ : (λ, φ) = (Λ(s), κ(s)) : s > 0 = ((π/L)2 , ∞). Finally, if (λ, φ) is a solution of (9.5) φ can be extended as a smooth 2L periodic function on the real line. When this has been done, let   RK = (K 2 λ, φ(Kx)) : (λ, φ) ∈ R . It is an easy matter to check that RK is a global branch of solutions bifurcating from (KL/π)2 , K ∈ N. Thus many qualitative features of the global bifurcation of solutions of (9.5), observed originally in the introduction, are a consequence of abstract considerations based on the theory of real analytic varieties and it is clear that the abstract method has much greater applicability. In the remaining chapters we give a substantial example to which the global theory makes a vital contribution.

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9.4 NOTES ON SOURCES The concept of global bifurcation has its origins in the work of Crandall and Rabinowitz [21] on ordinary differential equations, and was developed to include partial differential equations first by Rabinowitz [50] and then by Dancer [25] and Turner [64]. However, the approach for analytic operators is due to Dancer [26]. Some of the results given here appeared first in [13].

Chapter Ten Steady Periodic Water Waves Now we embark on a study of global bifurcation in a problem from classical hydrodynamics. A steady periodic irrotational water wave of infinite depth, with a free surface under gravity and without surface tension, is called a Stokes wave. The first mathematical treatment of this free-boundary problem is the local existence theory due to Levi-Civita [42] and Nekrasov [47]. Although a breakthrough at the time, this is now recognised as an example of bifurcation from a simple eigenvalue. The first global mathematical treatment is due to Krasov’skii [40], who obtained the existence of waves of all slopes from zero up to, but not including, π/6. However he did not show that they formed a connected set. That contribution was due to Keady and Norbury [36], who used the global topological bifurcation theory of Rabinowitz [50], as adapted for operator equations in cones by Dancer [25], to obtain the existence of a global connected set of Stokes waves. Here we take matters further and show that there is a global curve of solutions which has a local analytic re-parameterization at every point and which connects the bifurcation point representing zero flow (no wave) to a Stokes wave of extreme form [57]. The present account is based on a formulation of the Stokes-wave problem as a pseudo-differential operator equation due originally to Babenko [8] (see [63] for further background) and draws upon theory developed in [4, 11, 12, 13, 44]; see also [14, 53, 62, 63]. We begin with a description of the physical problem and a derivation of the equation that is the focus of our study.

10.1 EULER EQUATIONS  and a pressure field P in The incompressible Euler equations for a velocity field U a force field F are  · ∇)U  = ∇P + F , ∇ · U  = 0.  t + (U U In the next two sections we derive equations for steady irrotational water waves of infinite depth, with gravity but without surface tension. Steady Euler Equations We begin with a standard elliptic boundary-value problem on an unbounded domain.

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D EFINITION 10.1.1 A 2π-periodic function u : R → R is said to be H¨older continuous with exponent α ∈ (0, 1), written u ∈ C α , if

u C α =

sup

|u(x)| +

x∈[−π,π]

|u(x) − u(y)| < ∞. |x − y|α x, y∈[−π,π] sup

When u is k-times continuously differentiable on (−π, π) and the kth derivative has continuous extension to [−π, π] which is in C α , we say that u ∈ C k,α . Note that a function u which is Lipschitz continuous on [−π, π] need not be in C 1 , the space of continuously differentiable functions on [−π, π]; instead we write u ∈ C 0,1 when u is Lipschitz continuous. In this notation, C k , k ∈ N0 , denotes the space of 2πperiodic, k-times-continuously differentiable functions on R. This notation has an obvious extension to functions u defined on subsets U of Rm , C k,α (U ), C k,α (U ) etc. Note, for future reference that, as a consequence of the Ascoli-Arzel`a theorem 2.7.1, C k,α , α ∈ (0, 1), is compactly embedded in C k , k ∈ N0 . Suppose w is a real-valued, 2π-periodic, even C 2,α function of a real variable and that a curve S and a set Ω are defined by S := {(x, w(x)) : x ∈ R} and Ω = {(x, y) : y < w(x)}. For a real parameter c consider the boundary-value problem ∆ψˆ = 0 in Ω, ψˆ = 0 on S; ψˆ is 2π-periodic and even in x; ˆ y) − (0, c) → 0 as y → −∞. ∇ψ(x, For any c > 0 this problem has a unique solution which is real-analytic in Ω and all its derivatives and second derivatives have continuous extensions from Ω onto the boundary of Ω. To see this, a solution may be obtained by minimizing the functional
ˆ y) − (0, c)|2 dydx |∇ψ(x, Ω∩((−π,π)×R) 1,2 (Ω) which are 2π-periodic and even in x with over the set of all functions in Wloc zero trace on S. Then the Phrag`emen-Lindel¨of principle (the maximum principle for harmonic functions on unbounded domains) gives that the solution is unique. The regularity of the solution to the boundary-value problem obtained in this way is ensured by standard theory [33, §6.4] which yields that ψˆ ∈ C 2,α (Ω). Since ψˆ = 0 on S and ψˆ → −∞ as y → −∞, the harmonic function ψˆ is negative everywhere on Ω. It therefore attains its maximum on Ω at every point of S. Hence, by the boundary-point lemma,

∂ ψˆ > 0 on S. ∂y

STEADY PERIODIC WATER WAVES

129

ˆ Now note that the function ∂ ψ/∂y is a harmonic function on Ω which tends to ˆ c > 0 as y → −∞. The maximum principle therefore implies that ∂ ψ/∂y > 0 ˆ y) we infer that everywhere on Ω. From the evenness and periodicity of ψ(·, ∂ ψˆ = 0 at x = 0, ±π. ∂x It follows from the implicit function theorem that, for any α < 0, the set {(x, y) ∈ ˆ y) = α} is the graph of a smooth (in fact real-analytic) function Yα which Ω : ψ(x, gives y as a function of x. Define a velocity field  (x, y, t) = (u(x, y, t), v(x, y, t)) = (ψˆy , −ψˆx ). U  is independent of time t and that div U  = ∇·U  = 0. In other words, Note that U   U is a stationary and solenoidal. Let Fg = (0, −g) = ∇(−gy) be the (constant) gravitational force field acting vertically downwards and define a scalar pressure field in Ω by ˆ y)|2 + gy. P (x, y) = 12 |∇ψ(x,  satisfies the Euler equations for a two-dimensional incompressible flow Then U under gravity:  · ∇)U  = ∇P + Fg , ∇ · U  = 0.  t + (U U  = ∇∧U  = 0 , it is also irrotational. From the fact that ψˆ = 0 on S, Since curl U and from the behaviour of ∇ψˆ at infinite depth, we have that 0 = ∇ψˆ · (1, wx ) = ψˆx + ψˆy wx on S,  → (c, 0) as y → −∞. U ˆ it is tangent to the curve S. Now  is everywhere perpendicular to ∇ψ, Since U  and a define time-dependent domains Ωt with boundaries St , a velocity field V pressure P as follows: Ωt = {(x, y) : (x + ct, y) ∈ Ω}, η(x, t) = w(x + ct), St = {(x, y) : (x + ct, y) ∈ ∂Ω} = {(x, y) ∈ R2 : y = η(x, t)},  (x, y, t) = U  (x + ct, y, t) − (c, 0), P(x, y, t) = P (x + ct, y). V Since   t + (V  .∇(x,y) )V  }  x + (U  · ∇(x,y) ))U  − cU  x ] {V = [c U (x,y,t) (x+ct,y)  = {∇P + Fg }(x+ct,y) = {∇(x,y) P + Fg }(x,y,t) ,  , P define a steady solution of the Euler equations. Remember that S, and hence V St , was specified at the outset and so we should think of St as a rigid boundary

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at time t which is moving horizontally from right to left with constant velocity c on the surface of an infinitely wide, infinitely deep, ocean under gravity. The  and P which satisfy the Euler equations, and flow it generates is described by V  V (x, y, t) → 0 as y → −∞. Steady Symmetric Water Waves If S and c are such that P is constant on S, the rigid moving boundary is not needed to maintain the motion since the pressure P at the surface St is in equilibrium with constant atmospheric pressure. Therefore steady symmetric periodic water waves are described by solutions of a boundary-value problem in which the domain Ω (described by a 2π-periodic even C 2,α function w), the function ψˆ and the parameter c > 0 are the unknowns: ∆ψˆ = 0 in Ω; ˆ w(x)) = 0 for all x ∈ R; ψ(x,

(10.1a) (10.1b)

ˆ y) → (0, c) as y → −∞; ∇ψ(x, ˆ ˆ y) = ψ(x ˆ + 2π, y), (x, y) ∈ Ω; ψ(−x, y) = ψ(x, 1 ˆ w(x))|2 + gw(x) ≡ constant, x ∈ R. |∇ψ(x, 2

(10.1c) (10.1d) (10.1e)

We have seen equations (10.1a)-(10.1d) already; the new ingredient here is (10.1e), which expresses the requirement that the pressure on the free surface is constant. Dimensionless Variables Let ψˆ = cψ. Then ∆ψ = 0 in Ω; ψ(x, w(x)) = 0 for all x ∈ R;

(10.2a) (10.2b)

∇ψ(x, y) → (0, 1) as y → −∞;

(10.2c)

ψ(−x, y) = ψ(x, y) = ψ(x + 2π, y), (x, y) ∈ Ω; 1 2

(10.2d)

|∇ψ(x, w(x))| + λw(x) ≡ , x ∈ R, λ = g/c . 2

1 2

2

(10.2e)

There is no loss of generality in taking the constant on the right side of (10.2e) to be a half, since this can always be achieved by relocating the origin in the y direction, an operation which has no effect on the other equations. The parameter λ is a dimensionless parameter in the water-wave problem, the square of the Froude number. Slight Generalization For future reference consider briefly the more general problem in which the surface S, which is 2π-periodic and even in x, is given in parametric form by S = {(X(t), Y (t)) : t ∈ R},

131

STEADY PERIODIC WATER WAVES

where t → (X(t), (Y (t)) is a globally injective, C 2,α function, and (10.2b) and (10.2e) are satisfied on S: ψ(X(t), Y (t)) = 0 and 12 |∇ψ(X(t), Y (t)))|2 + λY (t) ≡ 12 , t ∈ R. L EMMA 10.1.2 Suppose (X, Y ) is a C 2,α -function of t with X
(t)2 + Y
(t)2 > 0 and X
(t) ≥ 0 on R. Then X
(t) > 0 on R. Proof. Since (X(t), Y (t)) is a C 2,α function of t, we know [33, §6.4] that ψ and all its first and second derivatives are continuous on Ω. Suppose that X
(t0 ) = 0. Then the first boundary condition gives ψy (X(t0 ), Y (t0 )) = 0 and Y
(t0 )2 ψxx (X(t0 ), Y (t0 )) = ψx (X(t0 ), Y (t0 )) X

(t0 ).

(10.3)

Now P = 12 |∇ψ|2 + λy is a sub-harmonic function on Ω which is constant on S and tends to −∞ as y → −∞. By the Hopf boundary-point lemma its outward normal derivative on S is everywhere strictly positive. Since ψ < 0 on Ω, the outward normal is parallel to ∇ψ on the boundary and so 0 < ∇ψ(X(t0 , Y (t0 )) · ∇P (X(t0 ), Y (t0 )) = ψx (X(t0 ), Y (t0 )) Px (X(t0 ), Y (t0 )) = ψx ((X(t0 ), Y (t0 ))2 ψxx (X(t0 ), Y (t0 )) With (10.3), this gives that X

(t0 ) = 0. Since X
(t) ≥ 0 and X
(t0 ) = 0 it follows that X

(t0 ) = 0, which is a contradiction and the proof is complete. Trivial Solution For all λ > 0 the system (10.2) has a solution w ≡ 0, ψ(x, y) = y, S = {(x, 0) : x ∈ R}, Ω = R × (−∞, 0),  = (c, 0) of the Euler equations and which corresponds to the constant solution U uniform parallel flow in a horizontal direction. This is a trivial solution.

10.2 ONE-DIMENSIONAL FORMULATION To tackle the existence questions for non-trivial solutions using bifurcation theory we first need to re-formulate (10.2) as a nonlinear equation in the form F (λ, w) = 0, where w : R → R is 2π-periodic. Let −φ be the harmonic conjugate of ψ in (10.2) so that φ + iψ is holomorphic in Ω. Both ψ and φ are in C 2,α (Ω). Since ψ is even, ψx (x, y) = 0 for all (x, y) ∈ Ω with x ∈ {0, ±π} and the Cauchy-Riemann equations ensure that we may normalize φ so that φ is odd and φ(±π, y) is independent of y. Since
π
π φ(π, y) − φ(−π, y) = φx (x, y)dx = ψy (x, y)dx → 2π −π

−π

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CHAPTER 10

as y → −∞, φ(±π, y) = ±π for all (±π, y) ∈ Ω and φ(0, y) = 0 for all (0, y) ∈ Ω. Therefore, since ψy = 0 on Ω, a symmetric solution of the steady water-wave problem gives rise to a conformal bijection φ + iψ from Ω onto R := R × (−∞, 0) and from Ωπ := Ω ∩ {(−π, π) × R} onto Rπ := (−π, π) × (−∞, 0). Let Sπ = Ωπ ∩ S. When composed with an exponential bijection this gives a conformal bijection ζ, defined on Ωπ by   ζ(x + iy) = exp − i(φ(x, y) + iψ(x, y)) , which maps Ωπ onto D \ {ζ ∈ R : ζ ≤ 0}, the open unit disc cut along the nonpositive real axis (the point at infinity in Ωπ is mapped to the origin). Let Z denote its inverse, from D \ {ζ ∈ R : ζ ≤ 0} into the complex z-plane, z = x + iy. From the boundary conditions satisfied by φ and ψ, and from the behaviour of φ + iψ as y → −∞, it follows that the function Z(ζ) − i log ζ, ζ ∈ D \ {ζ ∈ R : ζ ≤ 0}, can be extended to the non-positive real axis to yield a complex-analytic function on D. (Here log is the usual branch of logarithm which is real on the positive real axis.) Therefore Z can be written in the form ∞



  Z(ζ) =i log ζ + ak ζ k = i log ζ + f (ζ) ,

(10.4)

k=0

say. The evenness of ψ with respect to x means that f is real when ζ is real and therefore all the coefficients ak are real. Let points in the complex ζ-plane be identified by polar coordinates ζ = reit (t no longer denotes time.) Then Z(ζ) = i log r − t +

∞ 

  ak rk i cos kt − sin kt .

(10.5)

k=0

It is clear that Ωπ , the image of Z on D, is bounded by the vertical lines x = ±π and the curve Sπ given parametrically by x = −t −

∞  k=1

ak sin kt, y =

∞ 

ak cos kt, t ∈ [−π, π].

(10.6)

k=0

Let us start again and suppose that f is a given holomorphic function on D which is real on the real axis. Let Z be given in terms of f by (10.4). Suppose that Z is a bijection onto its range Ωπ and define Φ + iΨ on Ωπ by Φ(Z(ζ)) + iΨ(Z(ζ)) = i log ζ = i log r − t

(10.7)

so that Ψ ≡ 0 on Sπ . We now ask if this defines a solution of (10.2). Since Im Z → −∞ corresponds to r → 0, we find from (10.4) and (10.7) that ∇Φ(x + iy) → (1, 0)

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STEADY PERIODIC WATER WAVES

as y → −∞. It is therefore automatic that when Ψ is defined on Ωπ using (10.7) and the mapping f in (10.4), Ψ is harmonic and even with respect to x on Ωπ , Ψ ≡ 0 on Sπ and ∇Ψ → (0, 1) as y → −∞. For Ψ to give a solution of the water-wave problem the only question remaining is whether 1 2

|∇Ψ|2 + λy ≡

1 2

on Sπ ?

(10.8)

This requirement can be written as a further condition to be satisfied by f in (10.4) or, equivalently, by the coefficients {ak } in (10.5). Since (Φ + iΨ)(Z(ζ)) = i log ζ,  d  dZ i Φ + iΨ (Z(ζ)) = dz dζ ζ and hence 2  |∇Ψ(Z(ζ))|2 = | Φ + iΨ)
(Z(ζ)) =

1 |ζ|2 |Z
(ζ)|2

.

In particular, when |ζ| = 1, so that Z(ζ) ∈ Sπ and ζ = eit , we have |∇Ψ(Z(ζ))|2 =

1 |Z
(ζ)|2

,

where Z
(ζ) =

   i i +i 1+ kak ζ k−1 = kak ζ k . ζ ζ ∞

∞

k=1

k=1

Therefore (10.8) implies that at the point ζ = eit on the unit circle, 1 |∇Ψ(Z(ζ))|2 =    ∞ k 2 1 + k=1 kak ζ ∞ ∞ −1    = (1 + kak cos kt)2 + ( kak sin kt)2 . k=1

k=1

To write this in a convenient notation, define an operator C on L2 [−π, π] by C(1) = 0, C(sin kt) = − cos kt, C(cos kt) = sin kt, k ≥ 1, extended by linearity and continuity. Note from the Riesz-Fischer theorem that, as an operator on L2 [−π, π], C ≤ 1 and that C 2 u = −u + [u], where [u] denotes the mean of a function u ∈ L2 [−π, π]. The bounded linear operator C is the well known conjugation operator [69] from classical harmonic analysis.

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Suppose that w  ∈ L2 [−π, π] is an even function with w
= dw/dt ∈ L2 [−π, π] ∞ and Fourier series k=0 ak cos kt. Then w(−π) = w(π) and 1 + Cw
(t) = 1 +

∞ 

kak cos kt,

(10.9)

k=1

|∇Ψ(Z(eit ))|2 =

1 , (1 + Cw
(t))2 + w
(t)2

(10.10)

and, according to (10.6), Sπ = {(t + Cw(t), w(t)) : t ∈ [−π, π]}.

(10.11)

Therefore the constant-pressure condition takes the form (1 − 2λw(t)){w
(t)2 + (1 + Cw
(t))2 } = 1, for almost all t ∈ [−π, π]. (10.12) Henceforth we will focus on (10.12). The argument which follows (10.6) shows that if w ∈ C 2,α is even and such that Z defined by (10.5) is globally injective and (10.12) holds then there exists a symmetric Stokes wave with profile given by   (t + Cw(t), w(t)) : t ∈ R . Conjugation Operator To proceed we will need some standard operator theory. Everything in this section is proved in [9], [51] or [69]. Complex Analysis In complex notation Ceint = −i sgn n eint , n ∈ Z,

(10.13)

with the convention that sgn 0 = 0. Let u be a 2π-periodic, smooth, real-valued function with  u ˆ(n)eint , t ∈ [−π, π], u ˆ(−n) = u ˆ(n), u(t) = n∈Z

=u ˆ(0) +



u ˆ(n)eint +

n∈N



u ˆ(n)eint

n∈N

∞ 

 =u ˆ(0) + Re 2 u ˆ(n)eint . n=1

Therefore u is the restriction (u(t) = U (eit )) to the unit circle S 1 of the real part U of a complex holomorphic function f on the unit disc D in C, where  f (z) = u ˆ(0) + 2 u ˆ(n)z n = U (z) + iV (z), |z| ≤ 1, n∈N

135

STEADY PERIODIC WATER WAVES

U and V are real and V (0) = 0. It follows that    V (eit ) = Im f (eit ) = −i u ˆ(n)eint − u ˆ(n)eint n∈N

n∈N

   u ˆ(n)eint − u ˆ(−n)e−int = −i n∈N

n∈N

  sgn n u ˆ(n)eint = Cu(t). = −i n∈Z

Thus the conjugation operation gives the restriction to S 1 of the imaginary part of a complex holomorphic function f on the closed unit disc when the restriction to the boundary of Re f is u and Im f (0) = 0. Further, by the Cauchy-Riemann equations in polar coordinates on the unit disc, ∂U  Cu
(t) =  , ∂r eit where ∆U = 0 on D, U (eit ) = u(t). The above discussion is rigorous when the functions in question are smooth on the closed unit disc or on the unit circle. When u is square integrable, the theory is only a little more subtle. Functional Analysis For u ∈ L2 [−π, π], Cu (defined above in terms of Fourier coefficients) is given pointwise almost everywhere by the singular integral formula
π 1 u(s)ds PV , (10.14) Cu(t) = 1 2π tan −π 2 (t − s) where P V denotes a Cauchy principal value integral. Formula (10.14) defines Cu pointwise almost everywhere for u ∈ L1 [−π, π] and, although C maps neither L1 [−π, π] nor L∞ [−π, π] into itself, we have the following. T HEOREM 10.2.1 (M. Riesz) C : Lp [−π, π] → Lp [−π, π] is a bounded linear operator for all p ∈ (1, ∞). T HEOREM 10.2.2 all α ∈ (0, 1).

(Privalov) C : C α → C α is a bounded linear operator for

T HEOREM 10.2.3 Suppose that u ∈ L2 [−π, π]. Then there exists a holomorphic function f defined on the unit disc such that, in L2 [−π, π] and pointwise for almost all t ∈ [−π, π], lim f (reit ) = u(t) + iCu(t).

r→1

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CHAPTER 10

Suppose that u, w ∈ L2 [−π, π], α, β ∈ R and f, g are holomorphic in the unit disc with lim f (reit ) = u(t) + i(α + Cu(t)),

r→1

lim g(reit ) = w(t) + i(β + Cw(t)),

r→1

in L2 [−π, π] and pointwise for almost all t ∈ [−π, π]. Then there exists γ ∈ R and a function v ∈ L1 [−π, π] such that Cv ∈ L1 [−π, π] and, in L1 [−π, π] and pointwise for almost all t ∈ [−π, π], lim f (reit )g(reit ) = v(t) + i(γ + Cv(t)).

r→1

 In particular, if f g S 1 is imaginary, then v = 0 and f g = iγ on D for some real constant γ. Suppose that u is even and 2π-periodic. Let u ˜(t) = u(t + π). Then u ˜ is even, 2π-periodic and + Cu ˜(t) = (Cu)(t + π) = Cu(t).

(10.15)

L EMMA 10.2.4 Let D denote the unit disc in C and suppose that f : D → C has the following properties: f ∈ C 2,α (D) is holomorphic in D; f |∂D is injective so that f (∂D) is a simple closed Jordan curve Γ. Let Ω be the interior of Γ, the bounded component of C \ Γ. Then (a) Ω = f (D) and (b) f : D → Ω is a bijection. Proof. Since f is non-constant on ∂D, it is non-constant on D. Since it is holomorphic on D, f maps open sets to open sets. Therefore ∂(f (D)) ⊂ f (∂D) = Γ. If f (D) ⊂ Ω, then there exists a path in C \ Ω jointing x0 ∈ f (D) to infinity. Therefore there is a point of ∂(f (D)) which is not in Ω and so not in Γ. This contradiction proves that f (D) ⊂ Ω. Since f (D) is open, f (D) ⊂ Ω. Now suppose that f (D) = Ω. Then there is a path in Ω joining a point of f (D) to a point of Ω not in f (D). Once again there is a point of ∂(f (D)) not in Γ, which is a contradiction. This proves part (a). To prove (b), note again that if z1 ∈ ∂D and z2 ∈ D, then f (z1 ) = f (z2 ), as in the argument for part (a). Suppose then that z1 , z2 ∈ D, z1 = z2 and f (z1 ) = f (z2 ) = ω ∈ Ω. Let r ∈ (0, 1) be such that |zi | < r, i = 1, 2. Then by [59, 3.4, page 115],
f
(z)dz 1 ≥ 2, 2πi ∂Dr f (z) − ω where Dr = {z ∈ D : |z| < r}. Since |f (z) − ω| is bounded away from 0 on ∂D, and f
∈ L1 (∂D), it follows (see, for example, [7, Thm. 6.6, page 100]) that
1 f
(z)dz ≥ 2. 2πi ∂D f (z) − ω

137

STEADY PERIODIC WATER WAVES

However, 1 2πi


∂D

1 f
(z)dz = f (z) − ω 2πi


Γ

dζ = 1, ζ −ω

since ω ∈ Ω, the interior of Γ. This contradiction implies (b) and the lemma is proven.

10.3 MAIN EQUATION To prove the existence of non-trivial solutions of equation (10.12) we replace it with another which is more convenient from many viewpoints. The relevant observation is the following. D EFINITION 10.3.1 Let Y denote the space of 2π-periodic absolutely continuous even functions u : R → R with u
∈ L2 [−π, π], endowed with the usual norm

u 2Y = u 2L2 [−π,π] + u
2L2 [−π,π] . Let X and Z be the spaces of even and odd functions in L2 [−π, π]. T HEOREM 10.3.2

Suppose that w ∈ Y is a solution of Cw
= λ{w + wCw
+ C(ww
)}.

(10.16)

(a) w satisfies (10.12). (b) 1 − 2λw > 0 on [−π, π] if and only if w
∈ L3 [−π, π]. (c) If w ∈ C 2,α , 1 − 2λw > 0 and 1 + Cw
≥ 0 on [−π, π], then Z in (10.7) is injective and the argument following (10.6) means that the solution w of (10.16) gives rise to a symmetric Stokes wave with profile   S = (t + Cw(t), w(t)) : t ∈ R where 1 + Cw
> 0 on [−π, π]. Proof. (a) First re-write (10.16) in the form   (1 − 2λw)(1 + Cw
) + C (1 − 2λw)w
= 1

(10.17)

and apply C to both sides to obtain   (1 − 2λw)w
= C (1 − 2λw)(1 + Cw
) so that Cu = (1 − 2λw)w
where u = (1 − 2λw)(1 + Cw
). Therefore, by Theorem 10.2.3, there exists a holomorphic function U on D with  U S 1 = u + iCu = (1 − 2λw){1 + Cw
+ iw
} = i(1 − 2λw){w
− i(1 + Cw
)}.

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CHAPTER 10

 By the same theorem there is a holomorphic function W on D such that W S 1 = w
+ i(1 + Cw
). Note also that the product U W has the property that    2 U W S 1 = i(1 − 2λw) w
+ (1 + Cw
)2 ∈ iR. By the last part of Theorem 10.2.3, U W is constant on D and, by Cauchy’s integral formula and (10.17),
π
π 1 1 U (0) = (u + iCu)dt = u dt 2π −π 2π −π
π 1 (1 − 2λw)(1 + Cw
)dt = 1. = 2π −π Similarly W (0) = i, and therefore U W ≡ i on D. This identity restricted to S 1 gives (1 − 2λw){w
+ (1 + Cw
)2 } ≡ 1, 2

which is (10.12). (b) Suppose that w ∈ Y satisfies (10.16) and that 1 − 2λw > 0 everywhere. Since w is continuous there exists > 0 such that 1 − 2λw ≥ . Since, by Theorem 10.3.2, w satisfies (10.12) it is immediate that w
∈ L∞ [−π, π] ⊂ L3 [−π, π]. Now suppose that w
∈ L3 [−π, π]. The M. Riesz theorem implies that Cw
∈ L3 [−π, π] also. Suppose, seeking a contradiction, that 1 − 2λw(a) = 0 for some a ∈ [−π, π]. Then, by H¨older’s inequality, 
t    |1 − 2λw(t)| = 2λ w
(s)ds ≤ 2λ w
L3 [−π,π] |t − a|2/3 . a

Hence, because w satisfies (10.12), w
(t)2 + (1 + Cw
(t))2 ≥

1 . 2λ w
L3 [−π,π] |t − a|2/3

/ L1 [−π, π], w
+ (1 + Cw
)2 ∈ / L3/2 [−π, π], which is a contraSince |t − a|−1 ∈ diction. (c) From our hypotheses and (10.12), t → (t+Cw(t), w(t)), t ∈ R is an injective function with H¨older continuous second derivatives, by Privalov’s theorem 10.2.2. It also follows from classical potential theory that ψ is C 2 on Ω. That Z is a global injection follows immediately from this observation and Lemma 10.2.4. Hence the argument following (10.6) gives the existence of a Stokes wave with symmetric free surface, in the notation of Lemma 10.1.2, given by 2

S = {(X(t), Y (t)) : t ∈ R} = (t + Cw(t), w(t) : t ∈ R}. Lemma 10.1.2 implies that 1 + Cw
> 0 on [−π, π] and the proof is complete. Some obvious features of (10.16) are the following.

139

STEADY PERIODIC WATER WAVES

(i) It is the Euler-Lagrange equation (Definition 3.4.2) of the functional
J (w) =

π

−π



 wCw
− λw2 (1 + Cw
) dt, w ∈ Y.

(ii) It is a quadratic equation (with no higher order terms) the non-zero solutions of which give rise to exact solutions of the steady periodic water-wave problem (without approximation). (iii) It has the trivial solution w = 0 for all λ. Linearizing with respect to w about the trivial solution yields the self-adjoint eigenvalue problem Cw
= λw, w ∈ Y. Since Y consists of even functions, the linearized problem has a complete set of eigenfunctions {cos nt}n∈N∪{0} , the corresponding set of eigenvalues being N ∪ {0}. (iv) It can be re-written (1 − 2λw)Cw
= λ{w − Q(w)},

(10.18)

  C (1 − 2λw)w
= λ{w + Q(w)}

(10.19)

or as

where Q(u) = uCu
− C(uu
). (See §10.5 for properties of Q(u).) (v) It can be re-written as G(λ, w) = 0, where G : (0, ∞) × Y → X is defined by G(λ, w) = Cw
− λ{w + wCw
+ C(ww
)}.

(10.20)

For (λ, u) ∈ U where U = {(λ, w) ∈ (0, ∞) × Y : 1 − 2λw > 0},

(10.21)

∂w G[(λ, w)] : Y → X is Fredholm with index 0. (This is proved in §10.5.) All these features are favourable; an awkward aspect of (10.16) is the involvement of the conjugation operator which is non-local and there is no obvious analogue of the maximum principle. To extract a priori bounds on solutions of (10.16) we introduce the classical equation for Stokes waves.

140

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10.4 A PRIORI BOUNDS AND NEKRASOV’S EQUATION Let w be an even C 2,α function such that (10.16), and hence (10.12), is satisfied. By Theorem 10.2.3 there exists a holomorphic function U on the unit disc D with U (eit ) = 1 + Cw
(t) − iw
(t). Suppose in addition that 1 + Cw
> 0 on [−π, π]. Then U is nowhere zero on D and its logarithm is well-defined. In polar coordinates let log U (reit ) = log |U (reit | + iΘ(reit ) where Θ(eit ) = ϑ(t). By (10.12), log U (eit ) = log |U (eit )| + iϑ(t) = − 12 log(1 − 2λw(t)) + iϑ(t), where ϑ(t) ∈ (− 12 π, 12 π) and tan ϑ(t) =

−w
(t) , t ∈ [−π, π]. 1 + Cw
(t)

By Privalov’s theorem 10.2.2, ϑ
∈ C 1,α since w ∈ C 2,α . By the Cauchy-Riemann equations,  ∂Θ  ∂ 1 −λw
(t)  it = 2 log(1 − 2λw(t)) = ∂r e ∂t 1 − 2λw(t) = λ (1 − 2λw(t))−3/2 sin ϑ(t). , On the other hand sin ϑ(t) = −w
(t) 1 − 2λw(t) gives
t   3λ ν + sin ϑ(s) ds = (1 − 2λw(t))3/2 .

(10.22)

0

The new parameter ν, which is related to (λ, w) in (10.16) by ν = (1 − 2λw(0))3/2 3λ, is called Nekrasov’s parameter, Θ(eit ) = ϑ(t) and

∂Θ  sin ϑ(t)  =   *t ∂r eit 3 ν + 0 sin ϑ(s) ds

(10.23)

Since w is even, ϑ is odd. Note that the parameter λ does not appear in (10.23). Moreover the favourable properties (i)–(v) are lost. Nevertheless it leads to the following observation which will be useful later. L EMMA 10.4.1 (10.16) with

Suppose w is a non-constant, even, C 2,α function which satisfies

1 + Cw
≥ 0 and 1 + Cw
+ |w
| > 0 on [−π, π], w
≤ 0 on (0, π). Then 1 + Cw
> 0 on [−π, π], w
< 0 on (0, π), w

(0) < 0 < w

(π) and ϑ
(0) > 0 > ϑ
(π).

141

STEADY PERIODIC WATER WAVES

Proof. That 1 + Cw
> 0 on [−π, π] is immediate from Theorem 10.3.2 (c) and Lemma 10.1.2. Hence it suffices to prove that 0 ≡ ϑ ≥ 0 on (0, π) implies that ϑ > 0 on (0, π). Since Θ is continuous on D and odd in t, it follows that Θ is non-negative on the boundary of the upper half disc D+ = {reit : t ∈ [0, π], r ∈ [0, 1]}. If ϑ(t0 ) = 0, t0 ∈ (0, π), then, by the maximum principle, Θhas a minimum on D+ at eit0 and, by the Hopf boundary-point lemma, ∂Θ/∂r eit0 < 0. However, by (10.23), this is a contradiction which shows that w
< 0 on (0, π). Since ∂Θ/∂r = 0 on the real axis in D and (∂Θ/∂r)(eit ) > 0, t ∈ (0, π), by (10.23), the maximum principle gives that the harmonic function r∂Θ/∂r is + positive on D + and takes its minimum on D at every point of ∂D on the real axis. The Hopf boundary-point lemma gives that ∂2Θ ∂2Θ (r) > 0 > (−r), r ∈ (0, 1). ∂t∂r ∂t∂r   Since ∂Θ/∂t (±r) → 0 as r → 0 and w
(0) = w
(π) = 0, −w

(0) ∂Θ = sec2 Θ(1) (1) > 0 1 + Cw
(0) ∂t −w

(π) ∂Θ 0 > sec2 Θ(−1) (−1) = . ∂t 1 + Cw
(π) This completes the proof. Therefore, when w is even and 1 + Cw
≥ 0, ϑ is odd and 0 < ϑ < 12 π on (0, π). So, by elementary Fourier-series methods, the boundary-value problem (10.23) is equivalent to an integral equation for the odd function ϑ ∈ C 1,α . On [0, π] it must satisfy
1 π sin ϑ(s) *s ϑ(t) = ds, (10.24a) K(t, s) 3 0 ν + 0 sin ϑ(τ )dτ 0 < ϑ(t) < 12 π, t ∈ (0, π),

(10.24b)

where ν > 0 and K(t, s) =

  ∞  sin 12 (s + t)  1 2  sin kt sin ks  > 0. = log  π k π sin 12 (s − t) 

(10.25)

k=1

R EMARK 10.4.2 The sine-series expression for K is immediate from the Fourierseries method used to derive (10.24) from (10.23). For later use we now derive the close-form formula (10.25) which gives the positivity of K. Recall (see, for example, [38, Ch. XII, §55, V]) that for z ∈ C with |z| ≤ 1 and z = 1, − log(1 − z) =

∞  zk k=1

k

,

142

CHAPTER 10

where log here means the principal logarithm. In particular, for x ∈ (0, 2π), ∞  cos kx

k

k=1

= − log |2 sin 12 x| and

∞  sin kx k=1

k

=

1 (π − x). 2

(10.26)

Hence, for s, t ∈ [0, π], s = t, ∞ N   sin kt sin ks cos k(t − s) − cos k(t + s) = lim N →∞ k 2k k=1 k=1   1 1  sin 2 (t + s)  = log   > 0, 2 sin 12 (t − s)

which gives (10.25). Now by (10.26)
π
π sin kt π cot 12 t dt = −2 cos kt log |2 sin 12 t|dt = . k k 0 0 Therefore


π

K(s, t) cot t dt = 2 1 2

0

∞  sin ks k=1

k

= π − s, s ∈ (0, π)

(10.27)

and (10.32) below follows from the symmetry of K. Suppose that ν > 0 and ϑ satisfy (10.24). Then ϑ(0) = ϑ(π) = 0 and either ϑ is identically zero or ϑ(x) > 0 on (0, π) because of the positivity of the kernel K. Moreover, multiplying the equation by sin x and integrating over (0, π) using (10.25) yields that
π
1 π sin ϑ(t) sin t dt ϑ(x) sin x dx = * 3 0 ν + t sin ϑ(w)dw 0 0
π 1 ϑ(x) sin x dx, < 3ν 0 which implies 0 < ν < 1/3. (10.28) L EMMA 10.4.3 Suppose that ϑ satisfies (10.24) on [0, π]. Then there exists c > 0 (independent of ϑ, ν and t) such that
t ν+ sin ϑ(τ )dτ ≥ c t, t ∈ [0, π]. 0 −1

Proof. Note that x √sin x is decreasing on (0, π) and x−β sin x is increasing on (0, π/3) for β = π/3 3. Hence 2 ϑ(t) ≤ sin ϑ(t) ≤ ϑ(t), t ∈ (0, 12 π) π

143

STEADY PERIODIC WATER WAVES

and, for s, t ∈ (0, π/3),        sin 12 (s + t)  s + t s + t π     .  √ log  ≤ log  ≤ log  s − t sin 12 (s − t)  s − t 3 3 Therefore, by (10.24), for x ∈ (0, π/6)


2x x

ϑ(t) dt t
π

  
2x 1  sin 12 (s + t)   sin ϑ(s)   dt ds *s log  t sin 12 (s − t)  0 ν + 0 sin ϑ(τ )dτ x  
2x 
2x 1 s + t  ϑ(s) 2   dt ds *s √ log  t s − t 9π 3 x ν + 0 ϑ(τ )dτ x  
2x 
2x/s 1 1 + u  ϑ(s) 2   du ds *s √ log  u 1 − u 9π 3 x ν + 0 ϑ(τ )dτ x/s
2x ϑ(s) log 3 *s √ ds 18π 3 x ν + 0 ϑ(τ )dτ * 2x  ϑ(τ )dτ  log 3 x* √ log 1 + , x ν + 0 ϑ(τ )dτ 18π 3

1 = 3π ≥ = ≥ =

since the interval [x/s, 2x/s] has length at least a half when s ∈ [x, 2x] and so


2x/s

x/s

    1 1 + u 1 + u  1 1 1     = log 3. log  du ≥ min log  u 1 − u 2 u∈[ 12 ,2] u 1 − u 4

Since ϑ < 12 π,


2x

x

ϑ(θ) dt ≤ 12 π log 2, t

and it follows from the above inequality that * 2x ϑ(τ )dτ x* 1+ x ν + 0 ϑ(τ )dτ is bounded, by M , say, independent of ϑ, ν and x. Since there exists K > 0 such that for all m ∈ [0, M ], log(1 + m) ≥ Km,


2x

x

 x * 2x ϑ(τ ) dτ   * 2x ϑ(τ )dτ  ϑ(θ) τ x* x* ≥K , dt ≥ K x x t ν + 0 ϑ(τ )dτ ν + 0 ϑ(τ )dτ

where K > 0 changes at each step but is independent of ν, ϑ and x. This proves the result for t ∈ [0, π/6] and the lemma follows.

144

CHAPTER 10

T HEOREM 10.4.4

Let ϑ ∈ C 1,α satisfy (10.24). Then 0 > t ϑ
(t) − ϑ(t), t ∈ (0, π), ϑ(t) < π/3, t ∈ [0, π], 0 0 on (0, π), ϑ
(0) > 0 > ϑ
(π) and ϑ
(t) = ϑ
(0) + N (t) where |N (t)| ≤ const tα . Hence for any A > 1, tϑ
(t) − Aϑ(t) < 0 for t in a left neighbourhood of π and, for t in a right neighbourhood of 0,
t

tϑ (t) − Aϑ(t) = (1 − A)tϑ (0) + tN (t) − A N (s)ds < −C t 0

if A > 1 where C > 0 is a constant. Suppose that the first inequality of the theorem is false. One possibility is that tϑ
(t) − ϑ(t) ≤ 0 on [−π, π] and t0 ϑ
(t0 ) − ϑ(t0 ) = 0, t0 ∈ (0, π). If this is not the case then 1 < A = inf{a ≥ 1 : tϑ
(t) − aϑ(t) ≤ 0, t ∈ (0, π)} and, from the behaviour of tϑ
(t) − ϑ(t) in a neighbourhood of 0 and π, there exists t0 ∈ (0, 1) with t0 ϑ
(t0 ) − Aϑ(t0 ) = 0. Hence, in all cases, there exists A ≥ 1 and t0 ∈ (0, π) such that tϑ
(t) − Aϑ(t) ≤ 0 on [−π, π] and t0 ϑ
(t0 ) − Aϑ(t0 ) = 0. As noted in the proof of Lemma 10.4.1, Θ > 0 on D+ , and hence ∂Θ  ≤0, 0 0. On the other hand, an integration of (10.16) yields that
π
π w(t)dt = −λ w(t)Cw
(t)dt < 0. −π

−π

146

CHAPTER 10

Hence there is a point in (0, π) where w is zero. The Lp [−π, π]-bound on w
follows from (10.12), (10.22) and Lemma 10.4.3. This in turn yields the bound on w and the proof is complete.

10.5 WEAK SOLUTIONS ARE CLASSICAL For w to satisfy (10.16) or (10.12) it is sufficient for w to have a square-integrable derivative (or merely to be absolutely continuous). On the other hand, in §10.1 the profile (X(t), Y (t)), t ∈ R, of a steady wave has w ∈ C 2,α . Thus §10.2 contains a discussion of weak solutions (which we prefer because of the favourable properties (i)–(v) listed there) to the classical problem discussed in §10.1. We now show that provided 1 − 2λw > 0 every weak solution is a classical solution. The Operator Q Formula (10.13) for the conjugation operator leads to the observation that u → Cu
is an unbounded, densely defined, self-adjoint operator on L2 [−π, π] given in terms of Fourier coefficients by u ˆ(k) → |k|ˆ u(k), where u ˆ(k) denotes the k th Fourier coefficient of u, k ∈ Z. The Cauchy principle value formula for C leads to the following observation which plays a central rˆole in the Stokes-wave problem. For a periodic function u with u
∈ L2 [−π, π] let Q(u) denote the function Q(u)(t) = u(t)Cu
(t) − C(uu
)(t).

L EMMA 10.5.1 almost all t ∈ R

Suppose that u is 2π-periodic and u
∈ L2 [−π, π]. Then for

1 0 ≤ Q(u)(t) = 8π




π

−π

0

u(t) − u(s) sin 12 (t − s)

12 ds ≤

2
2

u L2 [−π,π] . π

(10.33)

Proof. Fix t ∈ [−π, π] and observe, from Hardy’s inequality [51] and the periodicity of u, that, as an almost-everywhere defined function of s, u(t) − u(s) 2 tan 12 (t − s) is in L2 [−π, π] with norm bounded by 2 u
L2 [−π,π] , independent of t. Therefore
π u(t) − u(s)
1

u(t)Cu (t) − C(uu )(t) = u (s)ds 2π −π tan 12 (t − s) 2 ≤ u
2L2 [−π,π] . π *s Let t ∈ R be such that s → t u
(x)dx is differentiable with respect to s at s = t. According to Lebesgue’s theorem [51] this set has full measure in R. For

147

STEADY PERIODIC WATER WAVES

almost all such t, u(t)Cu
(t) − C(uu
)(t) =

1 2π




π

−π

(u(t) − u(s))u
(s) cot( 12 (t − s))ds


s 2 d
cot( (t − s)) u (x)dx ds ds π t 12
π 0 u(t) − u(s) 1 ds ≥ 0. = 8π −π sin 12 (t − s)

−1 = 4π




π

1 2

The equality in (10.33) holds for all t ∈ R when u is 2π-periodic and infinitely differentiable. Therefore, for such functions, integrating both sides of (10.33) and Parseval’s identity gives 2π




|k||ˆ u(k)| =

−π

k∈Z

L EMMA 10.5.2 2 Q(u) ∈ C 1− p .

π

2

1 8π




π

0

−π

u(t) − u(s) sin 12 (t − s)

12 dsdt.

Suppose u
∈ Lp [−π, π] for some p with 2 < p < ∞. Then

Proof. Since u is periodic and Q commutes with translations, it will suffice to 2 show that |Q(u)(x) − Q(u)(0)| ≤ const |x|1− p , for all x with |x| < π/2, where the constant depends only on u
p and on p. Without loss of generality suppose x > 0. Let p1 + 1q = 1 and let I denote the set [−π, π] \ [−2x, 2x]. Then |Q(u)(x) − Q(u)(0)|   2 2
2x * x u
(t)dt
2x * 0 u
(t)dt x−y −y 1 1 dy + dy ≤ 8π −2x 8π −2x sin2 (y/2) sin2 (y/2)  * 2 * 2   x 0

  1
u (t)dt − u (t)dt x−y −y   dy  . + 2   8π I sin (y/2)   Since u
∈ Lp [−π, π], p > 2, it follows from H¨older’s inequality that


2x

−2x

* x

u
(t)dt x−y

2

sin2 (y/2)

≤ const u
2p |x| q −1 , 2




2x

dy + −2x

* 0

u
(t)dt −y sin2 (y/2)

2 dy

148

CHAPTER 10

where the constant depends only on p. Also,   2 * 2  
* x 0

  u (t)dt − u (t)dt x−y −y   dy  ≤ 2  I  sin (y/2)    *    * x * *0 0 x 
 u
(t)dt + −y u
(t)dt u
(t)dt − −y u
(t)dt   x−y x−y  dy  2  sin (y/2)  I   
|y| q1 * x u
(t)dt − * 0 u
(t)dt x−y −y
≤ const u p dy 2 sin (y/2) I  
|y| q1 * x u
(t)dt − * x−y u
(t)dt 0 −y dy = const u
p sin2 (y/2) I

≤ const

1

u
2p |x| q




|y| q −2 dy ≤ const u
2p |x| q −1 . 1

2

I

Thus |Q(u)(x) − Q(u)(0)| ≤ const u
2p |x| q −1 = const u
2p |x|1− p . 2

2

This completes the proof. L EMMA 10.5.3 α.

Suppose that u
∈ C α , α ∈ (0, 1). Then Q(u) ∈ C 1,δ , 0 < δ
0 for almost all x ∈ [−π, π] unless u ≡ constant. (b) Q(u) ∈ L∞ [−π, π]. (c) Q(u) ∈ C α for all α ∈ (0, 1) if u
∈ Lp [−π, π] for all p < ∞. (d) Q(u) ∈ C 1,α for all α ∈ (0, 1) if u
∈ C β for all β ∈ (0, 1). T HEOREM 10.5.4 C 2,α .

Suppose that (λ, w) ∈ U is a solution of (10.16). Then w ∈

Proof. This follows by a simple bootstrap argument. Assume that 1 − 2λw > 0 and w satisfies (10.16). Then (10.18) and (b) imply that Cw
=

λ(w − Q(w)) ∈ L∞ [−π, π] ⊂ Lp [−π, π] 1 − 2λw

(10.34)

for all p ≥ 1. Hence, by Riesz’s theorem, w
∈ Lp [−π, π], 1 < p < ∞, and so w ∈ C β for all β ∈ (0, 1), by H¨older’s inequality, and Q(w) ∈ C β for all β ∈ (0, 1) by (c). Therefore, by (10.34), Cw
∈ C β for all β ∈ (0, 1) from which it follows, by Privalov’s theorem, that w
∈ C β , β ∈ (0, 1). Therefore Q(w) ∈ C 1,α , by (d) and 1 − 2λw ∈ C 1,α . Therefore, by (10.34), Cw
∈ C 1,α for all α ∈ (0, 1). It now follows, by Privalov’s theorem, that w ∈ C 2,α for all α ∈ (0, 1). This completes the proof.

150

CHAPTER 10

It is clear that this procedure could be continued indefinitely to prove by induction that w can be extended as infinitely differentiable 2π-periodic function on R. Theorem 10.3.2 (b) shows that the hypothesis 1−2λw > 0 (strict inequality) everywhere is essential to this conclusion. R EMARK 10.5.5 If {(λk , wk )} is a sequence of solutions of (10.16) with the property that 1 − 2λwk (t) ≥ d > 0, t ∈ [−π, π] and {λk } is bounded, then (10.12) gives that {wk
} is bounded in L2 [−π, π] and hence, by the preceding argument, {wk } is bounded in C 2,α for all α ∈ (0, 1). Therefore, by the last sentence in Definition 10.1.1, {wk } is compact in C 2 . Fredholm Property We begin with a general observation. T HEOREM 10.5.6 Q is sequentially continuous from Y , with the weak topology, into Lp [−π, π], 1 < p < ∞, with the strong Lp -topology. Proof. Let {vn } be a sequence in Y which is weakly convergent to v. Then vn
∈ Z and vn
 v
in L2 [−π, π]. It will suffice to show for a subsequence that Q(vn ) → *t Q(v) in Lp [−π, π], 1 < p < ∞, as n → ∞. Let un (t) = 0 (vn
(s) − v
(s))ds. Since Q(un )(t) = Q(vn − v)(t) and since the integral of Cu is zero for any u ∈ L2 [−π, π],
π
π Q(vn − v)(t)dt = un (t)C(un
)(t)dt −π −π
π un (t)C(vn
− v
)(t)dt → 0 = −π

as n → ∞, because C is bounded on L2 [−π, π], vn
 v
and un → 0 in L2 [−π, π]. Since Q(vn − v) is non-negative, it follows that Q(vn − v) → 0 in L1 [−π, π] and so a subsequence converges pointwise almost everywhere. Let *t *t
v (t)dt v
(t)dt n t−s and φt (s) = t−s . φtn (s) = sin(s/2) sin(s/2) We have just shown for a subsequence that, for almost all t, φtn → φt in L2 [−π, π] as n → ∞, and hence that φtn L2 [−π,π] → φt L2 [−π,π] . Another way of saying this is that, for almost all t, Q(vn )(t) → Q(v)(t) as n → ∞. Since {Q(vn )} is bounded in L∞ [−π, π], by Lemma 10.5.1, the result of the theorem follows from the dominated convergence theorem. To show that the linearization of G at (λ, w) ∈ U is Fredholm with index 0 it suffices (Theorem 2.7.6) to decompose ∂w G[(λ, w)] as ∂w G[(λ, w)] = K[λ, w] + C[λ, w],

STEADY PERIODIC WATER WAVES

151

where K[λ, w] : Y → X is a homeomorphism and C[λ, w] : Y → X is compact. Since, for (λ, w) ∈ R × Y, G(λ, w) = (1 − 2λw)(Cw
+ w) + λQ(w) − (1 − 2λw + λ)w, differentiation gives ∂w G[(λ, w)]ϕ = (1 − 2λw)(Cϕ
+ ϕ)

 + λdQ[w]ϕ − 2λϕ(Cw
+ w) − λϕ + 4λwϕ − ϕ . (10.35) Since 1 − 2λw is everywhere positive and continuous, the first term on the right obviously represents a homeomorphism from Y to X. Since the nonlinear operator Q is compact from Y into X (Theorem 10.5.6), its Fr´echet derivative dQ[w] : Y → X is also compact (Lemma 3.1.12). Also since, by the Ascoli-Arzel`a theorem 2.7.1, Y is compactly embedded (see Example 2.7.3) in L∞ [−π, π], all the other terms in the second bracket of (10.35) denote compact linear operators from Y into X. This shows that ∂w G[(λ, w)] : Y → X is Fredholm with index 0 on U. Note that U is open in R × Y .

10.6 NOTES ON SOURCES For the maximum principle, the Hopf boundary-point lemma and the Phrag`emenLindel¨of principle, see [31] and [49]. See also the Notes on Sources for Chapter 11.

Chapter Eleven Global Existence of Stokes Waves

11.1 LOCAL BIFURCATION THEORY Here we study the existence of non-trivial solutions of equation (10.16) in a neighbourhood of the trivial solution w = 0 ∈ Y for values of λ > 0. This equation can be re-written as G(λ, w) = 0 where G, defined by (10.20), has the following properties. (A) G(λ, 0) = 0 for all λ > 0.  

(B) ∂w G[(λ, w)]ϕ = Cϕ ϕCw
+ C(wϕ)
.  − λϕ −
λ wCϕ +  (C) ∂λ G[(λ, w)]µ = µ w + wCw + C(ww
) . Since ∂w G : (0, ∞) × Y → L(Y, X) and ∂λ G : (0, ∞) × Y → L(R, X) are both continuous, we conclude (Lemma 3.1.7) that G : (0, ∞) × Y → X is continuously Fr´echet differentiable. In fact all its partial Fr´echet derivatives of all orders greater than three are zero everywhere. Thus G : (0, ∞) × Y → X is a real-analytic operator. Since G is continuously Fr´echet differentiable and ∂w G[(λ, 0)] is Fredholm with index zero, a necessary condition (see Example 8.1.1) for there to be bifurcation from the trivial solution at λ0 > is that ∂w G[(λ0 , 0)] : Y → X is not injective, equivalently Cϕ
− λ0 ϕ = 0 has a non-trivial solution in Y. Bifurcation from a Simple Eigenvalue Since ∂w G[(λ, 0)]ϕ = Cϕ
− λϕ = 0, ϕ ∈ Y \ {0}, if and only if n ∈ N0 and ϕ(t) = a cos nt, a ∈ R, all bifurcation points λ0 are non-negative integers. Now we will show that every non-negative integer is a bifurcation point. It is easily seen that λ0 = 0 is a bifurcation point since (10.16) has a solution (λ, w) = (0, c) for all constants c. Now suppose that λ0 = n ∈ N. We need only check the hypotheses of the theory of bifurcation from a simple eigenvalue. Note that / range ∂w G[(n, 0)], ker ∂w G[(n, 0)] = span {ϕn } and ϕn ∈

GLOBAL EXISTENCE OF STOKES WAVES

153

where ϕn (t) = cos nt. Thus every n ∈ N is a simple eigenvalue of A : Y → X, where Au = Cu
, and hence every n ∈ N0 is a bifurcation point for the equation G(λ, w) = 0. T HEOREM 11.1.1 Let n ∈ N0 . Then there is a neighbourhood On of (n, 0) in R × Y , > 0 and a unique real-analytic function (Λn , Φn ) : (− , ) → On with
π ϕn (t)Φn (s)(t)dt = 0, s ∈ (− , ), −π

such that (λ, w) ∈ On is a solution of (10.16) with w = 0 if and only in there exists s ∈ (− , ) \ {0} such that  (λ, w) = (Λn (s), s ϕn + Φn (s))) where (Λn (0), Φn (0)) = (n, 0). Although the bifurcation from λ0 = 0 was known from the elementary observation that constants are solutions when λ = 0, the theorem gives the additional information that in a neighbourhood of (0, 0) in R × Y there are no other solutions. Bifurcation from λ = 1 Let E denote the set of all non-trivial solutions (λ, w) ∈ (0, ∞) × Y of (10.16) and let Γ1 = {(Λ1 (s), s(ϕ1 + Φ1 (s))) : s ∈ (− , ) \ {0}} ⊂ E denote the curve of non-trivial solutions bifurcating from (1, 0). The theory so far predicts that in a neighbourhood of the bifurcation point (1, 0) there are no nontrivial solutions of (10.16) other than those on the bifurcating curve Γ1 . Let E1 denote the maximal connected subset in R × Y of E which contains Γ1 . Now observe that in a neighbourhood of (1, 0) the curve Γ1 is symmetric in R × Y about the λ-axis because (λ, w) ˜ ∈ E1 whenever (λ, w) ∈ Γ1 (see (10.15)) and, by the uniqueness of the function (Λ1 , Φ1 ), since ϕ 21 = −ϕ1 ,   (Λ1 (−s), (−s)(ϕ1 + Φ1 (−s))) = Λ1 (s), s(ϕ1 + Φ1 (s))∼ for all s ∈ (0, ). Hence Λ˙ 1 = (d/ds)Λ1 (s)|s=0 = 0. (Here ˙ denotes differentiation with respect to s at s = 0.) In other words the tangent to the bifurcating curve Γ1 is vertical in R × Y at the bifurcation point. We shall show presently that the whole curve Γ1 is not vertical, unlike the case of bifurcation from λ0 = 0. Let Γ+ 1 := {(Λ1 (s), s(ϕ1 + Φ1 (s))) : s ∈ (0, )}, Y1 = span{ϕ1 } and let Y2 denote the closure in Y of span{ϕk : k = 0, 2, 3, . . . }. Substituting (λ, w) = (Λ1 (s), s(ϕ1 + sΦ1 (s))) in (10.16) gives 0 = (1 − Λ1 (s))ϕ1 + CΦ1 (s)
− Λ1 (s)Φ1 (s) − sΛ1 (s){(ϕ1 + Φ1 (s))C(ϕ
1 + Φ1 (s)
) + C((ϕ1 + Φ1 (s))(ϕ1 + Φ1 (s))
)}. (11.1)

154

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A differentiation with respect to s at s = 0, using the fact that Λ˙ 1 = 0, gives ˙
− Φ˙ 1 = ϕ1 Cϕ
+ C(ϕ1 ϕ
) = ϕ2 + 1 ϕ2 = 1 + ϕ2 . CΦ 1 1 1 1 2 2 Therefore Φ˙1 = ϕ2 − 12 . Differentiating (11.1) again with respect to s at s = 0 gives ¨
− Φ¨1 − 2ϕ1 − 6ϕ3 . ¨ 1 ϕ1 + C Φ 0 = −Λ 1 Hence ¨ 1 = 3ϕ3 . ¨ 1 = −2 and Φ Λ Therefore Γ1 is a symmetric, sub-critical pitchfork bifurcation at (1, 0). Consider the eigenvalue problem for the linearization of (10.16) with respect to w at (Λ1 (s), ws ), where ws = s(ϕ1 + Φ1 (s)), ∂w G[(Λ1 (s), ws )]v = νv, v ∈ X.

(11.2)

In the notation of Proposition 8.4.2, F = −G (see (8.4) and (10.20)) and so ν = −µ. Therefore, for s sufficiently small, all solutions (ν, v) of (11.2) in a sufficiently small neighbourhood of (0, ϕ1 ) (up to a normalization of v) are given by a smooth mapping s → (νs , vs ) such that (ν0 , v0 ) = (0, ϕ1 ) and lim

s→0

sdΛ1 (s)/ds = 1. νs

Further, from the symmetry of Γ1 we may infer that νs is even in s and v−s = v2s . These conclusions may be summarized as follows:  ws := s(ϕ1 + Φ1 (s)) = sϕ1 + s2 (ϕ2 − 12 ) + 32 s3 ϕ3  Λ1 (s) = 1 − s2 + O(|s|4 ). (11.3)  2 νs = −2s as |s| → 0. For non-zero s with |s| sufficiently small, all but two of the eigenvalues of ∂w G[(Λ1 (s), ws )] are greater than 1/2, since that is true when s = 0 and the eigenvalues ν of (11.2) depend continuously on s. Of the two exceptions, one is close to −1 (because −1 is an eigenvalue of (11.2) when s = 0), and the other is negative by (11.3). Thus, when positive |s| is sufficiently small, equation (11.2) has exactly two negative eigenvalues and all the others exceed 1/2. We return briefly to this observation after we have defined Morse indices in §11.3: it says that near the bifurcation point λ = 1 all non-trivial solutions have Morse index 2.

11.2 GLOBAL BIFURCATION FROM λ = 1 Now we turn to a global analysis of the water-wave problem, exploiting its real analyticity and the Fredholm property §10.5. In the same notation, let E1+ denote

GLOBAL EXISTENCE OF STOKES WAVES

155

the closure of the maximal connected subset of E ∩ U in (0, ∞) × Y which contains Γ+ 1 (U is defined in (10.21)). The outcome real-analytic bifurcation theory is the following result on the existence of a global continuum in U of solutions of (10.16). Let S = {(λ, w) ∈ E1+ : ∂w G[(λ, w)] : Y → X is a homeomorphism.} T HEOREM 11.2.1 For equation (10.16) there exists a function h = (Λ, W ) : (0, ∞) → R × Y such that (i) h : (0, ∞) → S¯ ∩ U is continuous; (ii) h((0, 1)) ⊂ Γ+ 1 and lims0 h(s) = (1, 0); (iii) h is injective on h−1 (S); (iv) at all points s ∈ h−1 (S), h is real-analytic with Λ
(s) = 0; ¯ ⊂ (0, ∞) consists of isolated values; (v) the set h−1 (S\S) ¯ there exists an injective and con(vi) locally near every point s0 ∈ h−1 (S\S), tinuous re-parameterization of the set originally parameterized by h, such that s = γ(σ), |σ| ≤ 1, s0 = γ(0) and h ◦ γ is a real-analytic function (whose derivative may vanish only at 0). Let R = {h(s) : s ∈ [0, ∞)}. Since R ⊂ U and R is path-connected in Y , it follows from the remark at the end of §10.5 that R is path-connected in C k , for all k ∈ N. Let  W = (λ, w) ∈ R × C 2,α which satisfies (10.16) and  1 + Cw
> 0 on [−π, π], w
< 0 on (0, π), w

(0) < 0 < w

(π) . It is easy to see from the local bifurcation analysis of the preceding section that h(s) ∈ W for s ∈ (0, ) when > 0 is sufficiently small. However Lemma 10.4.1 implies that W ∩R is both open and closed in R and hence, by path-connectedness, R ⊂ W. From this and Theorem 10.4.4 it follows that if (λ, w) ∈ R, then 0 < k ≤ λ ≤ K < ∞ and |w| ≤ K, for some k, K > 0. Now it follows from (10.12) that if 1 − 2λw(0) ≥ δ > 0 for all (λ, w) ∈ R, then R is bounded in R × Y . It must therefore form a closed loop, by Theorem 9.1.1 (e), if 1 − 2λw(0) ≥ δ > 0 for all (λ, w) ∈ R. Now let K = {w ∈ Y : w
≤ 0 on (0, π)}. We have seen that R ⊂ K and K is a closed cone in Y . In order to show that R is not a closed loop in Y it suffices to confirm the hypotheses of Theorem 9.2.2. It has already been observed that hypotheses (a) and (b) hold, and (c) holds also because

156

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ξ0 = cos t ∈ K. Suppose that (λ, w) ∈ R. Then w ∈ C 2,α and ϑ ≤ π/3 on (0, π). Moreover all solutions of (10.16) close to (λ, w) in R × Y are close in R × C 2,α and the corresponding ϑ ≤ π/3. This proves that hypothesis (d) holds and so R is not a closed loop. We conclude that h(s) → ∂U as s → ∞, R lies in a bounded subset of [k, ∞) × W 1,p [−π, π], for some k > 0 and any p ∈ [1, 3) (Theorem 10.4.4), and 1 − 2λw(0) → 0 as s → ∞ when h(s) = (λ, w). Here W 1,p [−π, π] denotes the Banach space of 2π-periodic functions with weak derivative in Lp [−π, π]. It follows that every sequence in R has a subsequence which is convergent weakly in [k, ∞) × W 1,p [−π, π], 1 < p < 3, and strongly in [k, ∞) × C α , α ∈ (0, 2/3). Let 1

H = R \ R in C 2 . L EMMA 11.2.2 H ⊂ R × W 1,p [−π, π], 1 < p < 3 and every (λ, w) ∈ H satisfies (10.16) with 1 − 2λw(0) = 0. Moreover the set H is a compact, connected 1 subset of R × C 2 . Proof. Let {h(sk )} = {(λk , wk )} be a sequence in R which converges to (λ, w) ∈ 1 H in C 2 as k and sk → ∞. Extracting a subsequence if necessary, we may suppose that it also converges weakly to (λ, w) in W 1,p [−π, π]. Thus (λ, w) ∈ R × W 1,p [−π, π]. Multiplying (10.16) by a smooth 2π-periodic function φ and integrating before taking the limit gives
π   0= φ Cwk
− λk (wk + wk Cwk
+ C(wk wk
) dt −π
π   φ Cw
− λ(w + wCw
+ C(ww
) → −π

Cwk





→ Cw , wk
→ w
weakly in Y and wk → w uniformly on [−π, π] as because k → ∞. It follows that (λ, w) ∈ R × W 1,p [−π, π] satisfies (10.16). 1 It follows that H = R\R is compact in R×C 2 . To show that H is connected in 1 1 R × C 2 , suppose that it is not. Then there exists two disjoint compact (in R × C 2 ) subsets U, V such that both intersect H non-trivially, their union is H, and U ∩ V is empty. Let dist(U, V ) = d > 0. Suppose that u ∈ H ∩ U and v ∈ H ∩ V . Since u∈ / V and v ∈ / U , there exists h(tm ) with dist(h(tm ), U ) ≥ d/4, dist(h(tm ), V ) ≥ d/4 and tm → ∞ as m → ∞. The corresponding sequence {(λm , wm )} has a 1 subsequence which converges strongly in C 2 to a solution (λ, w) of (10.16) with 1 − 2λw(0) = 0 and dist((λ, w), U ) ≥ d/4, dist((λ, w), V ) ≥ d/4. This contradicts the fact that (λ, w) ∈ H and the proof is complete.

GLOBAL EXISTENCE OF STOKES WAVES

157

R EMARK 11.2.3 The set H consists of Stokes waves of greatest height which arise as the limit as s → ∞ of points on R. While it may not be a singleton, it is connected. Such a conclusion could not be derived using topological bifurcation theory because in that case there is no analogue of h(s) → ∂U as s → ∞, which is the key here. R EMARK 11.2.4

(Scaling) Let   Rk = (kλ, ϑ(kt)) : (λ, ϑ) ∈ R , k ∈ N.

It is easy to see that elements of Rk are solutions to (10.16) that give rise to Stokes waves of period 2π, but of minimal period 2π/k. The curve Rk represents a global curve bifurcating at the point (k, 0). It is clear from the real-analytic theory developed here that Rk is in fact the same curve as would be given by a study of the global bifurcation from a simple eigenvalue at λ = k.

11.3 GRADIENTS, MORSE INDEX AND BIFURCATION To finish, we give a brief sketch of how real-analytic bifurcation theory interacts with the gradient structure of (10.16) to conclude the existence of multiple secondary bifurcation points on the global branch. As the abstract theory upon which these conclusions are based is far removed from considerations of real-analyticity, we omit the proofs. It is worth emphasizing however that the existence of a path, not just a connected set, of solutions is essential. This is where the real-analyticity comes in. Recall the notion of gradient structure in a Banach space setting (Definitions 3.4.1 and 3.4.2). For certain operators with gradient structure, the Morse index of a solution is defined as follows. D EFINITION Suppose that Y is dense in a Hilbert space (X, ·, ·) and that G(λ, ·) is the gradient of a C 2 -functional g(λ, ·) with G(λ, y) = 0 and ∂y G[(λ, y)]− µι : Y → X a homeomorphism except for µ in a discrete set S(λ, y). For  µ ∈ S(λ, y) suppose that µι − ∂y G[(λ, y)] is a Fredholm operator of index zero. Then the Morse index M (λ, y) of the solution (λ, y) is the number of strictly negative eigenvalues of ∂y G[(λ, y0 )], counted  according to multiplicity. (The multiplicity of µ is dim ker µι − ∂y G[(λ, y)] .) D EFINITION We say that (λ0 , y0 ) is a bifurcation point for the equation G(λ, y) = 0 if there are two sequences {(λk , yk )}, {(λk , y2k )} of solutions of G(λ, y) = 0 with y2k = y2k for all k (note the same λk for both) converging to (λ0 , y0 ) in R × Y . The following theorem is due to Kielh¨ofer [37] and independently to Chow and Lauterbach [20]. P ROPOSITION Suppose that U ⊂ (0, ∞) × Y is an open set, G : U → X is C 2 and such that M (λ, y) is well-defined for every (λ, y) ∈ U with G(λ, y) = 0.

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Suppose also that for compact sets of solutions in U, the sets S(λ, y) are uniformly bounded below. Let S := {(λ(s), y(s)) : s ∈ (− , )} ⊂ U be a continuous curve of solutions to G(λ, y) = 0 such that 0 ∈ / S(λ(s), y(s)) for all s ∈ (− , ) \ {0} and that lim M (λ(s), y(s)) = lim M (λ(s), y(s)).

s0

s0

Then (λ(0), y(0)) is a bifurcation point. Morse Index of Stokes Waves Recall that (10.16) can be written as G(λ, w) = 0, where G : R × Y → X is given by G(λ, w) = Cw
− λ{w + ωCw
+ C(ww
)}. Let J : R × Y → R be defined by
1 π {wCw
− λw2 (1 + Cw
)}dt, λ > 0, w ∈ Y, J (λ, w) = 2 −π Then ∇J = G, G : R × Y → X and J : R × Y → R are real-analytic. If (λ, w) ∈ U and G(λ, w) = 0, then the Morse index M (λ, w) is well-defined. Since a change in Morse index results bifurcation points on the curve R, the following result is of profound significance. P ROPOSITION (Plotnikov [48]) If (λk , wk ) ∈ U is a sequence of solutions of (10.16) in Theorem 11.2.1 with 1 − 2λk wk (0) → 0 as k → ∞, then M (λk , wk ) → ∞. The following result follows immediately from the existence of a path of solutions given by the analytic global bifurcation theorem 11.2.1. (In the topological version of global bifurcation theory it is difficult, and in general not always possible, to be sure of the existence of such a path upon which secondary bifurcation points can be identified.) P ROPOSITION (Buffoni, Dancer and Toland [13]) There is an infinite set Σ of values of s > 0 at which h(s) ∈ S¯ \ S is a bifurcation point for the equation G(λ, w) = 0. The set {h(s) ∈ S¯ \ S, s ∈ Σ} ⊂ U is infinite. Concluding Remarks Numerical experiments suggest that in the physical domain the curve R gives a maximal connected set of Stokes waves of the fundamental period 2π, and that no Stokes waves of period 2π bifurcate from it. Nor does it self-intersect. If this is so (and there is no proof) then the bifurcation points given by the last proposition are all turning points (and R has infinitely many ‘S-shapes’, on a bifurcation diagram)

GLOBAL EXISTENCE OF STOKES WAVES

159

and, after scaling (Remark 11.2.4), there are analogous bifurcation points on each of the curves Rk , k ∈ N. Whatever the nature of these bifurcation points, their existence implies the existence of points on the curves Rq , for sufficiently large prime numbers q, where solutions of minimal period 2π/q bifurcate from Rq . These bifurcations cannot be re-scaled to give bifurcations on R. When these observations are interpreted in the physical domain, they correspond to period-multiplying bifurcations of Stokes waves. The physical waves which bifurcate have minimal period 2qπ and there are infinitely many period-multiplying bifurcation points for Stokes waves in the physical domain [13].

11.4 NOTES ON SOURCES Equation (10.16) is due to Babenko [8]. This approach to the Stokes wave problem appeared in [12, 13]. Related material is to be found in [11, 14, 48, 62]. An earlier version of global Stokes-wave theory is summarized in [60].

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Index

∂x F [(x0 , y0 )], 23 (ν0 , v0 ), 154 (νs , vs ), 154 Br (X), 53 C([0, 1], FN ) = C 0 ([0, 1], FN ), 12 C 1 (U, Y ), 25 C 2 (U, Y ), 29 C k -manifold, 38 C k (U, Y ), 42 C α , 128 k C ∞ (U, Y ) = ∩∞ k=1 C (U, Y ), 46 C 0,1 , 128 C k,α , 128 C k , 128 dF , 21 dF [x0 ]x, 21 Er , 53 Fr , 53 I, 15 Sk , 11 Tn [F ]x0 (x), 32 u(n) , 12 W 1,p [−π, π], 156 X, 137 X ∗ , 14 Y , 137 Y1 = span{ϕ1 }, Y2 , 153 Z, 137 [u], 133 U , 139 C, 11 F, 11 F-analytic functions, 45 F-analytic operators, 45 Im , 11 ker(A), 15 N, 116 R, 115 N, N0 , 11 ν, v, 154

B, 83 ⊕, 13 2 ∂x,y F [(x0 , y0 )], 30 ∂x22 F [(x0 , y0 )], 30 Re , 11 R, 11 range (A), 15 Br , 53 L(X, Y ), 15 Mp (X, Y ), 42 Mp,q (X, Y ; Z), 52 Q, 146 R+ , 114 S, 114 T , 114 var (U, G), 78 var (V × Cn−m , H), 82 Z, 11 analytic extension, 69 analytic implicit function theorem, 56 analytic inverse function theorem, 55 analytic manifold, 56 analytic mapping, operator, function, 45 Ascoli-Arzel`a theorem, 18 Banach spaces, real, complex, 12 bifurcation diagram, 4 bifurcation point, 1, 103 bifurcation theory, local, 103 bijective, 16 bilinear form, 29 boundary-point lemma, 127 bounded multilinear operator, 41 bounded linear functional, 14 bounded linear operator, 15 branch of solutions, 106 branch, closure, 83 branches, of analytic varieties, 82

168 Cauchy-Schwarz, 13 chain rule, 22 co-prime, 72 codimension, of linear space, 13 codimension, of analytic variety, 91 compact nonlinear operator, 26 compact operator, 18 completely continuous, 26 complexified subspaces, 86 component (in a metric space), 11 composition, analytic functions, 56 cones, 120 conjugate operator, 16 continuously Fr´echet differentiable, 25 convex, 11 Crandall-Rabinowitz, 105 degree of a polynomial, 70 determinants, 42 dimension of the germ of an analytic variety, 78 directional derivative, 21 discriminant of a polynomial, 73 distinguished arc, 116 dual space, 14 eigenvalue, 19 eigenvector, 19 embeddings, bounded, continuous, compact, 18 equivalent (sets) at a, 78 equivalent norms, 11 Euclid’s algorithm, 74 Euler equations, steady, 127 Euler-Bernoulli theory, 3 Euler-Lagrange equations, 35 factorization of polynomials, 70 Fa`a de Bruno formula, 44 Fredholm alternative, 19 Fredholm operator, 20 Fredholm operator, index, 19 Froude number, 130 Fr´echet derivative, 21 fundamental theorem of algebra, 70 generated, F-analytic variety generated by G on U , 78

INDEX

germs of analytic varieties, 78 germs of sets at a, 78 global bifurcation in cones, 120 global bifurcation theory, 114 gradient operator, 34 gradient structure, 35 gradient vectors, 32 gradient, infinite dimensional, 34 greatest common divisor, 71, 74 group of permutations, 11 Hahn-Banach corollary, 14 higher derivatives, 27 Hilbert space, 13 homeomorphism, 16 homogeneous functions, 42 Hopf boundary-point lemma, 127 H¨older continuous, 128 identity operator, 15 implicit function theorem, 35, 37 incompressible flow, 129 injective, 15 inner product space, 13 inverse function theorem, 35 irreducible germ, 80 irrotational, 129 Jacobian matrices, 32 linear functional, 14 linear functional analysis, 11 Lipschitz continuous, 23 Lyapunov-Schmidt reduction, 104 manifold, 38 manifold, analytic, 56 maximum principles, 127 mean value theorem, 22 mixed derivatives, 30 multilinear operator, 41 multiplicity of root of polynomials, 70 necessary condition for bifurcation, 103, 104 Nekrasov’s parameter, 140 Nemytskii operators, 25 Nemytskii operators, analytic, 50 Neumann series, 16

169

INDEX

non-singular solution, 116 normed linear space, 11 one-dimensional branches, 95 open mapping theorem, corollary, 16 partial differentiation, 23 periodic bifurcation, 112 permutations, permutation group, 11 perturbation of a simple eigenvalue, analytic, 57 perturbation of simple eigenvalue, 38 Phrag`emen-Lindel¨of principle, 127 power series, 45 principal coefficient, 70 principle of linearization, 5 product Banach, Hilbert space, 17 projection, 17 projection theorem, 76 Puiseux series, 95 real-on-real polynomial, 74 real-on-real-analytic function, 62 reduced problem, 104 remainder after n terms, 46 representation of C-analytic sets, 88 representatives of germs, 78 resultant, 72 resultant matrix, 72 Riemann extension theorem, 66 root of a polynomial, 70 roots, simple, multiple, 70 route of length N , 117

second Fr´echet derivative, 27 simple eigenvalue, 19 simple eigenvalue bifurcation, 109 simplification of a polynomial, 76 skew symmetric, 42 steady flow, 129 Stirling’s formula, 46 Stokes wave, 127 Stokes wave of greatest height, 8 structure of analytic varieties, 93 sufficient condition for bifurcation, 105 summable in norm, 12, 16, 46 surjective, 15 symmetric bilinear form, operator, 29 symmetric group, 11 symmetric multilinear operator, 42 Taylor polynomial, 32 Taylor series, 32 Taylor’s theorem, 31 topological direct sum, 13 topological complement, 13 total derivative, 25 transversality condition, 105, 106 trivial solutions, 3, 103 twice differentiable, 29 unit ball, open, closed, 12 Weierstrass analytic variety, 82 Weierstrass polynomial, 81 Weierstrass’s division theorem, 64 Weierstrass’s preparation theorem, 65